Article TOPREAL7, MML version 4.99.1005

:: TOPREAL7:th 1
theorem
for b1, b2 being Element of REAL
      st max(b1,b2) <= b1
   holds max(b1,b2) = b1;

:: TOPREAL7:th 2
theorem
for b1, b2, b3, b4 being Element of REAL holds
max(b1 + b3,b2 + b4) <= (max(b1,b2)) + max(b3,b4);

:: TOPREAL7:th 3
theorem
for b1, b2, b3, b4, b5, b6 being Element of REAL
      st b1 <= b2 + b3 & b4 <= b5 + b6
   holds max(b1,b4) <= (max(b2,b5)) + max(b3,b6);

:: TOPREAL7:th 4
theorem
for b1, b2 being Relation-like Function-like FinSequence-like set holds
dom b2 c= dom (b1 ^ b2);

:: TOPREAL7:th 5
theorem
for b1 being natural set
for b2, b3 being Relation-like Function-like FinSequence-like set
      st len b2 < b1 & b1 <= (len b2) + len b3
   holds b1 - len b2 in dom b3;

:: TOPREAL7:th 6
theorem
for b1, b2, b3, b4 being Relation-like Function-like FinSequence-like set
      st b1 ^ b2 = b3 ^ b4 & len b1 = len b3 & len b2 = len b4
   holds b1 = b3 & b2 = b4;

:: TOPREAL7:th 7
theorem
for b1, b2 being FinSequence of REAL
      st (len b1 = len b2 or dom b1 = dom b2)
   holds len (b1 + b2) = len b1 & dom (b1 + b2) = dom b1;

:: TOPREAL7:th 8
theorem
for b1, b2 being FinSequence of REAL
      st (len b1 = len b2 or dom b1 = dom b2)
   holds len (b1 - b2) = len b1 & dom (b1 - b2) = dom b1;

:: TOPREAL7:th 9
theorem
for b1 being FinSequence of REAL holds
   len b1 = len sqr b1 & dom b1 = dom sqr b1;

:: TOPREAL7:th 10
theorem
for b1 being FinSequence of REAL holds
   len b1 = len abs b1 & dom b1 = dom abs b1;

:: TOPREAL7:th 11
theorem
for b1, b2 being FinSequence of REAL holds
sqr (b1 ^ b2) = (sqr b1) ^ sqr b2;

:: TOPREAL7:th 12
theorem
for b1, b2 being FinSequence of REAL holds
abs (b1 ^ b2) = (abs b1) ^ abs b2;

:: TOPREAL7:th 13
theorem
for b1, b2, b3, b4 being FinSequence of REAL
      st len b1 = len b2 & len b3 = len b4
   holds sqr ((b1 ^ b3) + (b2 ^ b4)) = (sqr (b1 + b2)) ^ sqr (b3 + b4);

:: TOPREAL7:th 14
theorem
for b1, b2, b3, b4 being FinSequence of REAL
      st len b1 = len b2 & len b3 = len b4
   holds abs ((b1 ^ b3) + (b2 ^ b4)) = (abs (b1 + b2)) ^ abs (b3 + b4);

:: TOPREAL7:th 15
theorem
for b1, b2, b3, b4 being FinSequence of REAL
      st len b1 = len b2 & len b3 = len b4
   holds sqr ((b1 ^ b3) - (b2 ^ b4)) = (sqr (b1 - b2)) ^ sqr (b3 - b4);

:: TOPREAL7:th 16
theorem
for b1, b2, b3, b4 being FinSequence of REAL
      st len b1 = len b2 & len b3 = len b4
   holds abs ((b1 ^ b3) - (b2 ^ b4)) = (abs (b1 - b2)) ^ abs (b3 - b4);

:: TOPREAL7:th 17
theorem
for b1 being Element of NAT
for b2 being FinSequence of REAL
      st len b2 = b1
   holds b2 in the carrier of Euclid b1;

:: TOPREAL7:th 18
theorem
for b1 being Element of NAT
for b2 being FinSequence of REAL
      st len b2 = b1
   holds b2 in the carrier of TOP-REAL b1;

:: TOPREAL7:funcnot 1 => TOPREAL7:func 1
definition
  let a1, a2 be non empty MetrStruct;
  func max-Prod2(A1,A2) -> strict MetrStruct means
    the carrier of it = [:the carrier of a1,the carrier of a2:] &
     (for b1, b2 being Element of the carrier of it holds
     ex b3, b4 being Element of the carrier of a1 st
        ex b5, b6 being Element of the carrier of a2 st
           b1 = [b3,b5] &
            b2 = [b4,b6] &
            (the distance of it) .(b1,b2) = max((the distance of a1) .(b3,b4),(the distance of a2) .(b5,b6)));
end;

:: TOPREAL7:def 1
theorem
for b1, b2 being non empty MetrStruct
for b3 being strict MetrStruct holds
      b3 = max-Prod2(b1,b2)
   iff
      the carrier of b3 = [:the carrier of b1,the carrier of b2:] &
       (for b4, b5 being Element of the carrier of b3 holds
       ex b6, b7 being Element of the carrier of b1 st
          ex b8, b9 being Element of the carrier of b2 st
             b4 = [b6,b8] &
              b5 = [b7,b9] &
              (the distance of b3) .(b4,b5) = max((the distance of b1) .(b6,b7),(the distance of b2) .(b8,b9)));

:: TOPREAL7:funcreg 1
registration
  let a1, a2 be non empty MetrStruct;
  cluster max-Prod2(a1,a2) -> non empty strict;
end;

:: TOPREAL7:funcnot 2 => TOPREAL7:func 2
definition
  let a1, a2 be non empty MetrStruct;
  let a3 be Element of the carrier of a1;
  let a4 be Element of the carrier of a2;
  redefine func [a3, a4] -> Element of the carrier of max-Prod2(a1,a2);
end;

:: TOPREAL7:funcnot 3 => TOPREAL7:func 3
definition
  let a1, a2 be non empty MetrStruct;
  let a3 be Element of the carrier of max-Prod2(a1,a2);
  redefine func a3 `1 -> Element of the carrier of a1;
end;

:: TOPREAL7:funcnot 4 => TOPREAL7:func 4
definition
  let a1, a2 be non empty MetrStruct;
  let a3 be Element of the carrier of max-Prod2(a1,a2);
  redefine func a3 `2 -> Element of the carrier of a2;
end;

:: TOPREAL7:th 20
theorem
for b1, b2 being non empty MetrStruct
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2 holds
dist([b3,b5],[b4,b6]) = max(dist(b3,b4),dist(b5,b6));

:: TOPREAL7:th 21
theorem
for b1, b2 being non empty MetrStruct
for b3, b4 being Element of the carrier of max-Prod2(b1,b2) holds
dist(b3,b4) = max(dist(b3 `1,b4 `1),dist(b3 `2,b4 `2));

:: TOPREAL7:th 22
theorem
for b1, b2 being non empty Reflexive MetrStruct holds
max-Prod2(b1,b2) is Reflexive;

:: TOPREAL7:funcreg 2
registration
  let a1, a2 be non empty Reflexive MetrStruct;
  cluster max-Prod2(a1,a2) -> strict Reflexive;
end;

:: TOPREAL7:th 23
theorem
for b1, b2 being non empty symmetric MetrStruct holds
max-Prod2(b1,b2) is symmetric;

:: TOPREAL7:funcreg 3
registration
  let a1, a2 be non empty symmetric MetrStruct;
  cluster max-Prod2(a1,a2) -> strict symmetric;
end;

:: TOPREAL7:th 24
theorem
for b1, b2 being non empty triangle MetrStruct holds
max-Prod2(b1,b2) is triangle;

:: TOPREAL7:funcreg 4
registration
  let a1, a2 be non empty triangle MetrStruct;
  cluster max-Prod2(a1,a2) -> strict triangle;
end;

:: TOPREAL7:funcreg 5
registration
  let a1, a2 be non empty Reflexive discerning symmetric triangle MetrStruct;
  cluster max-Prod2(a1,a2) -> strict discerning;
end;

:: TOPREAL7:th 25
theorem
for b1, b2 being non empty Reflexive discerning symmetric triangle MetrStruct holds
[:TopSpaceMetr b1,TopSpaceMetr b2:] = TopSpaceMetr max-Prod2(b1,b2);

:: TOPREAL7:th 26
theorem
for b1, b2 being non empty Reflexive discerning symmetric triangle MetrStruct
      st the carrier of b1 = the carrier of b2 &
         (for b3 being Element of the carrier of b1
         for b4 being Element of the carrier of b2
         for b5 being Element of REAL
               st 0 < b5 & b3 = b4
            holds ex b6 being Element of REAL st
               0 < b6 & Ball(b4,b6) c= Ball(b3,b5)) &
         (for b3 being Element of the carrier of b1
         for b4 being Element of the carrier of b2
         for b5 being Element of REAL
               st 0 < b5 & b3 = b4
            holds ex b6 being Element of REAL st
               0 < b6 & Ball(b3,b6) c= Ball(b4,b5))
   holds TopSpaceMetr b1 = TopSpaceMetr b2;

:: TOPREAL7:th 27
theorem
for b1, b2 being Element of NAT holds
[:TOP-REAL b1,TOP-REAL b2:],TOP-REAL (b1 + b2) are_homeomorphic;