Article GROUP_6, MML version 4.99.1005

:: GROUP_6:th 1
theorem
for b1, b2 being non empty set
for b3 being Function-like quasi_total Relation of b1,b2 holds
      b3 is one-to-one
   iff
      for b4, b5 being Element of b1
            st b3 . b4 = b3 . b5
         holds b4 = b5;

:: GROUP_6:modenot 1 => GROUP_6:mode 1
definition
  let a1 be non empty Group-like associative multMagma;
  let a2 be Subgroup of a1;
  redefine mode Subgroup of a2 -> Subgroup of a1;
end;

:: GROUP_6:funcreg 1
registration
  let a1 be non empty Group-like associative multMagma;
  cluster (1). a1 -> strict normal;
end;

:: GROUP_6:funcreg 2
registration
  let a1 be non empty Group-like associative multMagma;
  cluster (Omega). a1 -> strict normal;
end;

:: GROUP_6:th 2
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Subgroup of b1
for b3 being Element of the carrier of b1
for b4 being Subgroup of b2
for b5 being Element of the carrier of b2
      st b5 = b3
   holds b5 * b4 = b3 * b4 & b4 * b5 = b4 * b3;

:: GROUP_6:th 3
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Subgroup of b1
for b3, b4 being Subgroup of b2 holds
b3 /\ b4 = b3 /\ b4;

:: GROUP_6:th 4
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of the carrier of b1 holds
(b2 * b3) * (b2 ") = b3 |^ (b2 ") &
 b2 * (b3 * (b2 ")) = b3 |^ (b2 ");

:: GROUP_6:th 6
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Subgroup of b1
for b3 being Element of the carrier of b1 holds
   (b3 * b2) * b2 = b3 * b2 & b3 * (b2 * b2) = b3 * b2 & (b2 * b2) * b3 = b2 * b3 & b2 * (b2 * b3) = b2 * b3;

:: GROUP_6:th 7
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of bool the carrier of b1
      st b2 = {[.b3,b4.] where b3 is Element of the carrier of b1, b4 is Element of the carrier of b1: TRUE}
   holds b1 ` = gr b2;

:: GROUP_6:th 8
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being strict Subgroup of b1 holds
      b1 ` is Subgroup of b2
   iff
      for b3, b4 being Element of the carrier of b1 holds
      [.b3,b4.] in b2;

:: GROUP_6:th 9
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Subgroup of b1
for b3 being normal Subgroup of b1
      st b3 is Subgroup of b2
   holds b3 is normal Subgroup of b2;

:: GROUP_6:funcnot 1 => GROUP_6:func 1
definition
  let a1 be non empty Group-like associative multMagma;
  let a2 be Subgroup of a1;
  let a3 be normal Subgroup of a1;
  assume multMagma(#the carrier of a3,the multF of a3#) is Subgroup of a2;
  func (A2,A3)`*` -> strict normal Subgroup of a2 equals
    multMagma(#the carrier of a3,the multF of a3#);
end;

:: GROUP_6:def 1
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Subgroup of b1
for b3 being normal Subgroup of b1
      st multMagma(#the carrier of b3,the multF of b3#) is Subgroup of b2
   holds (b2,b3)`*` = multMagma(#the carrier of b3,the multF of b3#);

:: GROUP_6:th 10
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Subgroup of b1
for b3 being normal Subgroup of b1 holds
   b2 /\ b3 is normal Subgroup of b2 & b3 /\ b2 is normal Subgroup of b2;

:: GROUP_6:funcnot 2 => GROUP_6:func 2
definition
  let a1 be non empty Group-like associative multMagma;
  let a2 be Subgroup of a1;
  let a3 be normal Subgroup of a1;
  redefine func a2 /\ a3 -> strict normal Subgroup of a2;
end;

:: GROUP_6:funcnot 3 => GROUP_6:func 3
definition
  let a1 be non empty Group-like associative multMagma;
  let a2 be normal Subgroup of a1;
  let a3 be Subgroup of a1;
  redefine func a2 /\ a3 -> strict normal Subgroup of a3;
end;

:: GROUP_6:attrnot 1 => STRUCT_0:attr 7
definition
  let a1 be 1-sorted;
  attr a1 is trivial means
    ex b1 being set st
       the carrier of a1 = {b1};
end;

:: GROUP_6:dfs 2
definiens
  let a1 be non empty 1-sorted;
To prove
     a1 is trivial
it is sufficient to prove
  thus ex b1 being set st
       the carrier of a1 = {b1};

:: GROUP_6:def 2
theorem
for b1 being non empty 1-sorted holds
      b1 is trivial
   iff
      ex b2 being set st
         the carrier of b1 = {b2};

:: GROUP_6:th 11
theorem
for b1 being non empty Group-like associative multMagma holds
   (1). b1 is trivial;

:: GROUP_6:funcreg 3
registration
  let a1 be non empty Group-like associative multMagma;
  cluster (1). a1 -> trivial strict;
end;

:: GROUP_6:exreg 1
registration
  cluster non empty trivial strict unital Group-like associative multMagma;
end;

:: GROUP_6:th 12
theorem
(for b1 being non empty trivial Group-like associative multMagma holds
    ord b1 = 1 & b1 is finite) &
 (for b1 being non empty finite Group-like associative multMagma
       st ord b1 = 1
    holds b1 is trivial);

:: GROUP_6:th 13
theorem
for b1 being non empty trivial strict Group-like associative multMagma holds
   (1). b1 = b1;

:: GROUP_6:funcnot 4 => GROUP_2:func 14
notation
  let a1 be non empty Group-like associative multMagma;
  let a2 be normal Subgroup of a1;
  synonym Cosets a2 for Left_Cosets a2;
end;

:: GROUP_6:funcreg 4
registration
  let a1 be non empty Group-like associative multMagma;
  let a2 be normal Subgroup of a1;
  cluster Left_Cosets a2 -> non empty;
end;

:: GROUP_6:th 15
theorem
for b1 being non empty Group-like associative multMagma
for b2 being set
for b3 being normal Subgroup of b1
      st b2 in Left_Cosets b3
   holds ex b4 being Element of the carrier of b1 st
      b2 = b4 * b3 & b2 = b3 * b4;

:: GROUP_6:th 16
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3 being normal Subgroup of b1 holds
   b2 * b3 in Left_Cosets b3 & b3 * b2 in Left_Cosets b3;

:: GROUP_6:th 17
theorem
for b1 being non empty Group-like associative multMagma
for b2 being set
for b3 being normal Subgroup of b1
      st b2 in Left_Cosets b3
   holds b2 is Element of bool the carrier of b1;

:: GROUP_6:th 18
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of bool the carrier of b1
for b4 being normal Subgroup of b1
      st b2 in Left_Cosets b4 & b3 in Left_Cosets b4
   holds b2 * b3 in Left_Cosets b4;

:: GROUP_6:funcnot 5 => GROUP_6:func 4
definition
  let a1 be non empty Group-like associative multMagma;
  let a2 be normal Subgroup of a1;
  func CosOp A2 -> Function-like quasi_total Relation of [:Left_Cosets a2,Left_Cosets a2:],Left_Cosets a2 means
    for b1, b2 being Element of Left_Cosets a2
    for b3, b4 being Element of bool the carrier of a1
          st b1 = b3 & b2 = b4
       holds it .(b1,b2) = b3 * b4;
end;

:: GROUP_6:def 4
theorem
for b1 being non empty Group-like associative multMagma
for b2 being normal Subgroup of b1
for b3 being Function-like quasi_total Relation of [:Left_Cosets b2,Left_Cosets b2:],Left_Cosets b2 holds
      b3 = CosOp b2
   iff
      for b4, b5 being Element of Left_Cosets b2
      for b6, b7 being Element of bool the carrier of b1
            st b4 = b6 & b5 = b7
         holds b3 .(b4,b5) = b6 * b7;

:: GROUP_6:funcnot 6 => GROUP_6:func 5
definition
  let a1 be non empty Group-like associative multMagma;
  let a2 be normal Subgroup of a1;
  func A1 ./. A2 -> multMagma equals
    multMagma(#Left_Cosets a2,CosOp a2#);
end;

:: GROUP_6:def 5
theorem
for b1 being non empty Group-like associative multMagma
for b2 being normal Subgroup of b1 holds
   b1 ./. b2 = multMagma(#Left_Cosets b2,CosOp b2#);

:: GROUP_6:funcreg 5
registration
  let a1 be non empty Group-like associative multMagma;
  let a2 be normal Subgroup of a1;
  cluster a1 ./. a2 -> non empty strict;
end;

:: GROUP_6:th 22
theorem
for b1 being non empty Group-like associative multMagma
for b2 being normal Subgroup of b1 holds
   the carrier of b1 ./. b2 = Left_Cosets b2;

:: GROUP_6:th 23
theorem
for b1 being non empty Group-like associative multMagma
for b2 being normal Subgroup of b1 holds
   the multF of b1 ./. b2 = CosOp b2;

:: GROUP_6:funcnot 7 => GROUP_6:func 6
definition
  let a1 be non empty Group-like associative multMagma;
  let a2 be normal Subgroup of a1;
  let a3 be Element of the carrier of a1 ./. a2;
  func @ A3 -> Element of bool the carrier of a1 equals
    a3;
end;

:: GROUP_6:def 6
theorem
for b1 being non empty Group-like associative multMagma
for b2 being normal Subgroup of b1
for b3 being Element of the carrier of b1 ./. b2 holds
   @ b3 = b3;

:: GROUP_6:th 24
theorem
for b1 being non empty Group-like associative multMagma
for b2 being normal Subgroup of b1
for b3, b4 being Element of the carrier of b1 ./. b2 holds
(@ b3) * @ b4 = b3 * b4;

:: GROUP_6:th 25
theorem
for b1 being non empty Group-like associative multMagma
for b2 being normal Subgroup of b1
for b3, b4 being Element of the carrier of b1 ./. b2 holds
@ (b3 * b4) = (@ b3) * @ b4;

:: GROUP_6:funcreg 6
registration
  let a1 be non empty Group-like associative multMagma;
  let a2 be normal Subgroup of a1;
  cluster a1 ./. a2 -> Group-like associative;
end;

:: GROUP_6:th 26
theorem
for b1 being non empty Group-like associative multMagma
for b2 being normal Subgroup of b1
for b3 being Element of the carrier of b1 ./. b2 holds
   ex b4 being Element of the carrier of b1 st
      b3 = b4 * b2 & b3 = b2 * b4;

:: GROUP_6:th 27
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3 being normal Subgroup of b1 holds
   b3 * b2 is Element of the carrier of b1 ./. b3 & b2 * b3 is Element of the carrier of b1 ./. b3 & carr b3 is Element of the carrier of b1 ./. b3;

:: GROUP_6:th 28
theorem
for b1 being non empty Group-like associative multMagma
for b2 being set
for b3 being normal Subgroup of b1 holds
      b2 in b1 ./. b3
   iff
      ex b4 being Element of the carrier of b1 st
         b2 = b4 * b3 & b2 = b3 * b4;

:: GROUP_6:th 29
theorem
for b1 being non empty Group-like associative multMagma
for b2 being normal Subgroup of b1 holds
   1_ (b1 ./. b2) = carr b2;

:: GROUP_6:th 30
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3 being normal Subgroup of b1
for b4 being Element of the carrier of b1 ./. b3
      st b4 = b2 * b3
   holds b4 " = b2 " * b3;

:: GROUP_6:th 32
theorem
for b1 being non empty Group-like associative multMagma
for b2 being normal Subgroup of b1 holds
   Ord (b1 ./. b2) = Index b2;

:: GROUP_6:th 33
theorem
for b1 being non empty Group-like associative multMagma
for b2 being normal Subgroup of b1
      st Left_Cosets b2 is finite
   holds Ord (b1 ./. b2) = index b2;

:: GROUP_6:th 34
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Subgroup of b1
for b3 being strict normal Subgroup of b1
      st b3 is Subgroup of b2
   holds b2 ./. ((b2,b3)`*`) is Subgroup of b1 ./. b3;

:: GROUP_6:th 35
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being strict normal Subgroup of b1
      st b3 is Subgroup of b2
   holds b2 ./. ((b2,b3)`*`) is normal Subgroup of b1 ./. b3;

:: GROUP_6:th 36
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being strict normal Subgroup of b1 holds
      b1 ./. b2 is non empty Group-like associative commutative multMagma
   iff
      b1 ` is Subgroup of b2;

:: GROUP_6:attrnot 2 => GROUP_6:attr 1
definition
  let a1, a2 be non empty multMagma;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is multiplicative means
    for b1, b2 being Element of the carrier of a1 holds
    a3 . (b1 * b2) = (a3 . b1) * (a3 . b2);
end;

:: GROUP_6:dfs 6
definiens
  let a1, a2 be non empty multMagma;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is multiplicative
it is sufficient to prove
  thus for b1, b2 being Element of the carrier of a1 holds
    a3 . (b1 * b2) = (a3 . b1) * (a3 . b2);

:: GROUP_6:def 7
theorem
for b1, b2 being non empty multMagma
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is multiplicative(b1, b2)
   iff
      for b4, b5 being Element of the carrier of b1 holds
      b3 . (b4 * b5) = (b3 . b4) * (b3 . b5);

:: GROUP_6:exreg 2
registration
  let a1, a2 be non empty Group-like associative multMagma;
  cluster Relation-like Function-like non empty total quasi_total multiplicative Relation of the carrier of a1,the carrier of a2;
end;

:: GROUP_6:modenot 2
definition
  let a1, a2 be non empty Group-like associative multMagma;
  mode Homomorphism of a1,a2 is Function-like quasi_total multiplicative Relation of the carrier of a1,the carrier of a2;
end;

:: GROUP_6:th 40
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2 holds
   b3 . 1_ b1 = 1_ b2;

:: GROUP_6:condreg 1
registration
  let a1, a2 be non empty Group-like associative multMagma;
  cluster Function-like quasi_total multiplicative -> unity-preserving (Relation of the carrier of a1,the carrier of a2);
end;

:: GROUP_6:th 41
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Element of the carrier of b1
for b4 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2 holds
   b4 . (b3 ") = (b4 . b3) ";

:: GROUP_6:th 42
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3, b4 being Element of the carrier of b1
for b5 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2 holds
   b5 . (b3 |^ b4) = (b5 . b3) |^ (b5 . b4);

:: GROUP_6:th 43
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3, b4 being Element of the carrier of b1
for b5 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2 holds
   b5 . [.b3,b4.] = [.b5 . b3,b5 . b4.];

:: GROUP_6:th 44
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3, b4, b5 being Element of the carrier of b1
for b6 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2 holds
   b6 . [.b3,b4,b5.] = [.b6 . b3,b6 . b4,b6 . b5.];

:: GROUP_6:th 45
theorem
for b1 being Element of NAT
for b2, b3 being non empty Group-like associative multMagma
for b4 being Element of the carrier of b2
for b5 being Function-like quasi_total multiplicative Relation of the carrier of b2,the carrier of b3 holds
   b5 . (b4 |^ b1) = (b5 . b4) |^ b1;

:: GROUP_6:th 46
theorem
for b1 being integer set
for b2, b3 being non empty Group-like associative multMagma
for b4 being Element of the carrier of b2
for b5 being Function-like quasi_total multiplicative Relation of the carrier of b2,the carrier of b3 holds
   b5 . (b4 |^ b1) = (b5 . b4) |^ b1;

:: GROUP_6:th 47
theorem
for b1 being non empty Group-like associative multMagma holds
   id the carrier of b1 is Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b1;

:: GROUP_6:th 48
theorem
for b1, b2, b3 being non empty Group-like associative multMagma
for b4 being Function-like quasi_total multiplicative Relation of the carrier of b2,the carrier of b1
for b5 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b3 holds
   b5 * b4 is Function-like quasi_total multiplicative Relation of the carrier of b2,the carrier of b3;

:: GROUP_6:funcnot 8 => GROUP_6:func 7
definition
  let a1, a2, a3 be non empty Group-like associative multMagma;
  let a4 be Function-like quasi_total multiplicative Relation of the carrier of a1,the carrier of a2;
  let a5 be Function-like quasi_total multiplicative Relation of the carrier of a2,the carrier of a3;
  redefine func a5 * a4 -> Function-like quasi_total multiplicative Relation of the carrier of a1,the carrier of a3;
end;

:: GROUP_6:funcnot 9 => GROUP_6:func 8
definition
  let a1, a2 be non empty Group-like associative multMagma;
  func 1:(A1,A2) -> Function-like quasi_total multiplicative Relation of the carrier of a1,the carrier of a2 means
    for b1 being Element of the carrier of a1 holds
       it . b1 = 1_ a2;
end;

:: GROUP_6:def 8
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2 holds
      b3 = 1:(b1,b2)
   iff
      for b4 being Element of the carrier of b1 holds
         b3 . b4 = 1_ b2;

:: GROUP_6:th 49
theorem
for b1, b2, b3 being non empty Group-like associative multMagma
for b4 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2
for b5 being Function-like quasi_total multiplicative Relation of the carrier of b2,the carrier of b3 holds
   b5 * 1:(b1,b2) = 1:(b1,b3) & (1:(b2,b3)) * b4 = 1:(b1,b3);

:: GROUP_6:funcnot 10 => GROUP_6:func 9
definition
  let a1 be non empty Group-like associative multMagma;
  let a2 be normal Subgroup of a1;
  func nat_hom A2 -> Function-like quasi_total multiplicative Relation of the carrier of a1,the carrier of a1 ./. a2 means
    for b1 being Element of the carrier of a1 holds
       it . b1 = b1 * a2;
end;

:: GROUP_6:def 9
theorem
for b1 being non empty Group-like associative multMagma
for b2 being normal Subgroup of b1
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b1 ./. b2 holds
      b3 = nat_hom b2
   iff
      for b4 being Element of the carrier of b1 holds
         b3 . b4 = b4 * b2;

:: GROUP_6:funcnot 11 => GROUP_6:func 10
definition
  let a1, a2 be non empty Group-like associative multMagma;
  let a3 be Function-like quasi_total multiplicative Relation of the carrier of a1,the carrier of a2;
  func Ker A3 -> strict Subgroup of a1 means
    the carrier of it = {b1 where b1 is Element of the carrier of a1: a3 . b1 = 1_ a2};
end;

:: GROUP_6:def 10
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2
for b4 being strict Subgroup of b1 holds
      b4 = Ker b3
   iff
      the carrier of b4 = {b5 where b5 is Element of the carrier of b1: b3 . b5 = 1_ b2};

:: GROUP_6:funcreg 7
registration
  let a1, a2 be non empty Group-like associative multMagma;
  let a3 be Function-like quasi_total multiplicative Relation of the carrier of a1,the carrier of a2;
  cluster Ker a3 -> strict normal;
end;

:: GROUP_6:th 50
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Element of the carrier of b1
for b4 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2 holds
      b3 in Ker b4
   iff
      b4 . b3 = 1_ b2;

:: GROUP_6:th 51
theorem
for b1, b2 being non empty strict Group-like associative multMagma holds
Ker 1:(b1,b2) = b1;

:: GROUP_6:th 52
theorem
for b1 being non empty Group-like associative multMagma
for b2 being strict normal Subgroup of b1 holds
   Ker nat_hom b2 = b2;

:: GROUP_6:funcnot 12 => GROUP_6:func 11
definition
  let a1, a2 be non empty Group-like associative multMagma;
  let a3 be Function-like quasi_total multiplicative Relation of the carrier of a1,the carrier of a2;
  func Image A3 -> strict Subgroup of a2 means
    the carrier of it = a3 .: the carrier of a1;
end;

:: GROUP_6:def 11
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2
for b4 being strict Subgroup of b2 holds
      b4 = Image b3
   iff
      the carrier of b4 = b3 .: the carrier of b1;

:: GROUP_6:th 53
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2 holds
   rng b3 = the carrier of Image b3;

:: GROUP_6:th 54
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being set
for b4 being Function-like quasi_total multiplicative Relation of the carrier of b2,the carrier of b1 holds
      b3 in Image b4
   iff
      ex b5 being Element of the carrier of b2 st
         b3 = b4 . b5;

:: GROUP_6:th 55
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2 holds
   Image b3 = gr rng b3;

:: GROUP_6:th 56
theorem
for b1, b2 being non empty Group-like associative multMagma holds
Image 1:(b1,b2) = (1). b2;

:: GROUP_6:th 57
theorem
for b1 being non empty Group-like associative multMagma
for b2 being normal Subgroup of b1 holds
   Image nat_hom b2 = b1 ./. b2;

:: GROUP_6:th 58
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b2,the carrier of b1 holds
   b3 is Function-like quasi_total multiplicative Relation of the carrier of b2,the carrier of Image b3;

:: GROUP_6:th 59
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b2,the carrier of b1
      st b2 is finite
   holds Image b3 is finite;

:: GROUP_6:funcreg 8
registration
  let a1 be non empty finite Group-like associative multMagma;
  let a2 be non empty Group-like associative multMagma;
  let a3 be Function-like quasi_total multiplicative Relation of the carrier of a1,the carrier of a2;
  cluster Image a3 -> finite strict;
end;

:: GROUP_6:th 60
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b2,the carrier of b1
      st b2 is non empty Group-like associative commutative multMagma
   holds Image b3 is commutative;

:: GROUP_6:th 61
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b2,the carrier of b1 holds
   Ord Image b3 c= Ord b2;

:: GROUP_6:th 62
theorem
for b1 being non empty finite Group-like associative multMagma
for b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2 holds
   ord Image b3 <= ord b1;

:: GROUP_6:attrnot 3 => GROUP_6:attr 2
definition
  let a1, a2 be non empty Group-like associative multMagma;
  let a3 be Function-like quasi_total multiplicative Relation of the carrier of a1,the carrier of a2;
  attr a3 is being_monomorphism means
    a3 is one-to-one;
end;

:: GROUP_6:dfs 11
definiens
  let a1, a2 be non empty Group-like associative multMagma;
  let a3 be Function-like quasi_total multiplicative Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is being_monomorphism
it is sufficient to prove
  thus a3 is one-to-one;

:: GROUP_6:def 12
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2 holds
      b3 is being_monomorphism(b1, b2)
   iff
      b3 is one-to-one;

:: GROUP_6:attrnot 4 => GROUP_6:attr 3
definition
  let a1, a2 be non empty Group-like associative multMagma;
  let a3 be Function-like quasi_total multiplicative Relation of the carrier of a1,the carrier of a2;
  attr a3 is being_epimorphism means
    rng a3 = the carrier of a2;
end;

:: GROUP_6:dfs 12
definiens
  let a1, a2 be non empty Group-like associative multMagma;
  let a3 be Function-like quasi_total multiplicative Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is being_epimorphism
it is sufficient to prove
  thus rng a3 = the carrier of a2;

:: GROUP_6:def 13
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2 holds
      b3 is being_epimorphism(b1, b2)
   iff
      rng b3 = the carrier of b2;

:: GROUP_6:prednot 1 => GROUP_6:attr 2
notation
  let a1, a2 be non empty Group-like associative multMagma;
  let a3 be Function-like quasi_total multiplicative Relation of the carrier of a1,the carrier of a2;
  synonym a3 is_monomorphism for being_monomorphism;
end;

:: GROUP_6:prednot 2 => GROUP_6:attr 3
notation
  let a1, a2 be non empty Group-like associative multMagma;
  let a3 be Function-like quasi_total multiplicative Relation of the carrier of a1,the carrier of a2;
  synonym a3 is_epimorphism for being_epimorphism;
end;

:: GROUP_6:th 63
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Element of the carrier of b2
for b4 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2
      st b4 is being_monomorphism(b1, b2) & b3 in Image b4
   holds b4 . (b4 " . b3) = b3;

:: GROUP_6:th 64
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Element of the carrier of b1
for b4 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2
      st b4 is being_monomorphism(b1, b2)
   holds b4 " . (b4 . b3) = b3;

:: GROUP_6:th 65
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b2,the carrier of b1
      st b3 is being_monomorphism(b2, b1)
   holds b3 " is Function-like quasi_total multiplicative Relation of the carrier of Image b3,the carrier of b2;

:: GROUP_6:th 66
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b2,the carrier of b1 holds
      b3 is being_monomorphism(b2, b1)
   iff
      Ker b3 = (1). b2;

:: GROUP_6:th 67
theorem
for b1 being non empty Group-like associative multMagma
for b2 being non empty strict Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2 holds
      b3 is being_epimorphism(b1, b2)
   iff
      Image b3 = b2;

:: GROUP_6:th 68
theorem
for b1 being non empty Group-like associative multMagma
for b2 being non empty strict Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2
   st b3 is being_epimorphism(b1, b2)
for b4 being Element of the carrier of b2 holds
   ex b5 being Element of the carrier of b1 st
      b3 . b5 = b4;

:: GROUP_6:th 69
theorem
for b1 being non empty Group-like associative multMagma
for b2 being normal Subgroup of b1 holds
   nat_hom b2 is being_epimorphism(b1, b1 ./. b2);

:: GROUP_6:attrnot 5 => GROUP_6:attr 4
definition
  let a1, a2 be non empty Group-like associative multMagma;
  let a3 be Function-like quasi_total multiplicative Relation of the carrier of a1,the carrier of a2;
  attr a3 is being_isomorphism means
    a3 is being_epimorphism(a1, a2) & a3 is being_monomorphism(a1, a2);
end;

:: GROUP_6:dfs 13
definiens
  let a1, a2 be non empty Group-like associative multMagma;
  let a3 be Function-like quasi_total multiplicative Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is being_isomorphism
it is sufficient to prove
  thus a3 is being_epimorphism(a1, a2) & a3 is being_monomorphism(a1, a2);

:: GROUP_6:def 14
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2 holds
      b3 is being_isomorphism(b1, b2)
   iff
      b3 is being_epimorphism(b1, b2) & b3 is being_monomorphism(b1, b2);

:: GROUP_6:prednot 3 => GROUP_6:attr 4
notation
  let a1, a2 be non empty Group-like associative multMagma;
  let a3 be Function-like quasi_total multiplicative Relation of the carrier of a1,the carrier of a2;
  synonym a3 is_isomorphism for being_isomorphism;
end;

:: GROUP_6:th 70
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2 holds
      b3 is being_isomorphism(b1, b2)
   iff
      rng b3 = the carrier of b2 & b3 is one-to-one;

:: GROUP_6:th 71
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2
      st b3 is being_isomorphism(b1, b2)
   holds dom b3 = the carrier of b1 & rng b3 = the carrier of b2;

:: GROUP_6:th 72
theorem
for b1 being non empty Group-like associative multMagma
for b2 being non empty strict Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2
      st b3 is being_isomorphism(b1, b2)
   holds b3 " is Function-like quasi_total multiplicative Relation of the carrier of b2,the carrier of b1;

:: GROUP_6:th 73
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b2,the carrier of b1
for b4 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2
      st b3 is being_isomorphism(b2, b1) & b4 = b3 "
   holds b4 is being_isomorphism(b1, b2);

:: GROUP_6:th 74
theorem
for b1, b2, b3 being non empty Group-like associative multMagma
for b4 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2
for b5 being Function-like quasi_total multiplicative Relation of the carrier of b2,the carrier of b3
      st b4 is being_isomorphism(b1, b2) & b5 is being_isomorphism(b2, b3)
   holds b5 * b4 is being_isomorphism(b1, b3);

:: GROUP_6:th 75
theorem
for b1 being non empty Group-like associative multMagma holds
   nat_hom (1). b1 is being_isomorphism(b1, b1 ./. (1). b1);

:: GROUP_6:prednot 4 => GROUP_6:pred 1
definition
  let a1, a2 be non empty Group-like associative multMagma;
  pred A1,A2 are_isomorphic means
    ex b1 being Function-like quasi_total multiplicative Relation of the carrier of a1,the carrier of a2 st
       b1 is being_isomorphism(a1, a2);
  reflexivity;
::  for a1 being non empty Group-like associative multMagma holds
::     a1,a1 are_isomorphic;
end;

:: GROUP_6:dfs 14
definiens
  let a1, a2 be non empty Group-like associative multMagma;
To prove
     a1,a2 are_isomorphic
it is sufficient to prove
  thus ex b1 being Function-like quasi_total multiplicative Relation of the carrier of a1,the carrier of a2 st
       b1 is being_isomorphism(a1, a2);

:: GROUP_6:def 15
theorem
for b1, b2 being non empty Group-like associative multMagma holds
   b1,b2 are_isomorphic
iff
   ex b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b2 st
      b3 is being_isomorphism(b1, b2);

:: GROUP_6:th 77
theorem
for b1, b2 being non empty strict Group-like associative multMagma
      st b1,b2 are_isomorphic
   holds b2,b1 are_isomorphic;

:: GROUP_6:prednot 5 => GROUP_6:pred 2
definition
  let a1, a2 be non empty strict Group-like associative multMagma;
  redefine pred a1,a2 are_isomorphic;
  symmetry;
::  for a1, a2 being non empty strict Group-like associative multMagma
::        st a1,a2 are_isomorphic
::     holds a2,a1 are_isomorphic;
  reflexivity;
::  for a1 being non empty strict Group-like associative multMagma holds
::     a1,a1 are_isomorphic;
end;

:: GROUP_6:th 78
theorem
for b1, b2, b3 being non empty Group-like associative multMagma
      st b1,b2 are_isomorphic & b2,b3 are_isomorphic
   holds b1,b3 are_isomorphic;

:: GROUP_6:th 79
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b2,the carrier of b1
      st b3 is being_monomorphism(b2, b1)
   holds b2,Image b3 are_isomorphic;

:: GROUP_6:th 80
theorem
for b1, b2 being non empty strict Group-like associative multMagma
      st b1 is trivial & b2 is trivial
   holds b1,b2 are_isomorphic;

:: GROUP_6:th 81
theorem
for b1, b2 being non empty Group-like associative multMagma holds
(1). b1,(1). b2 are_isomorphic;

:: GROUP_6:th 82
theorem
for b1 being non empty strict Group-like associative multMagma holds
   b1,b1 ./. (1). b1 are_isomorphic;

:: GROUP_6:th 83
theorem
for b1 being non empty Group-like associative multMagma holds
   b1 ./. (Omega). b1 is trivial;

:: GROUP_6:th 84
theorem
for b1, b2 being non empty strict Group-like associative multMagma
      st b1,b2 are_isomorphic
   holds Ord b1 = Ord b2;

:: GROUP_6:th 85
theorem
for b1, b2 being non empty Group-like associative multMagma
      st b1,b2 are_isomorphic & b1 is finite
   holds b2 is finite;

:: GROUP_6:th 86
theorem
for b1, b2 being non empty strict Group-like associative multMagma
      st b1,b2 are_isomorphic & b1 is finite
   holds Ord b1 = Ord b2;

:: GROUP_6:th 87
theorem
for b1 being non empty trivial strict Group-like associative multMagma
for b2 being non empty strict Group-like associative multMagma
      st b1,b2 are_isomorphic
   holds b2 is trivial;

:: GROUP_6:th 89
theorem
for b1 being non empty Group-like associative multMagma
for b2 being non empty strict Group-like associative multMagma
      st b1,b2 are_isomorphic & b1 is commutative
   holds b2 is commutative;

:: GROUP_6:th 90
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b2,the carrier of b1 holds
   b2 ./. Ker b3,Image b3 are_isomorphic;

:: GROUP_6:th 91
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Function-like quasi_total multiplicative Relation of the carrier of b2,the carrier of b1 holds
   ex b4 being Function-like quasi_total multiplicative Relation of the carrier of b2 ./. Ker b3,the carrier of Image b3 st
      b4 is being_isomorphism(b2 ./. Ker b3, Image b3) & b3 = b4 * nat_hom Ker b3;

:: GROUP_6:th 92
theorem
for b1 being non empty Group-like associative multMagma
for b2 being normal Subgroup of b1
for b3 being strict normal Subgroup of b1
for b4 being strict normal Subgroup of b1 ./. b3
      st b4 = b2 ./. ((b2,b3)`*`) & b3 is Subgroup of b2
   holds (b1 ./. b3) ./. b4,b1 ./. b2 are_isomorphic;

:: GROUP_6:th 93
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Subgroup of b1
for b3 being strict normal Subgroup of b1 holds
   (b2 "\/" b3) ./. ((b2 "\/" b3,b3)`*`),b2 ./. (b2 /\ b3) are_isomorphic;