Article PDIFF_1, MML version 4.99.1005

:: PDIFF_1:funcnot 1 => PDIFF_1:func 1
definition
  let a1, a2 be Element of NAT;
  func proj(A1,A2) -> Function-like quasi_total Relation of REAL a2,REAL means
    for b1 being Element of REAL a2 holds
       it . b1 = b1 . a1;
end;

:: PDIFF_1:def 1
theorem
for b1, b2 being Element of NAT
for b3 being Function-like quasi_total Relation of REAL b2,REAL holds
      b3 = proj(b1,b2)
   iff
      for b4 being Element of REAL b2 holds
         b3 . b4 = b4 . b1;

:: PDIFF_1:th 1
theorem
dom proj(1,1) = REAL 1 &
 rng proj(1,1) = REAL &
 (for b1 being Element of REAL holds
    (proj(1,1)) . <*b1*> = b1 &
     (proj(1,1)) " . b1 = <*b1*>);

:: PDIFF_1:th 2
theorem
(proj(1,1)) " is Function-like quasi_total Relation of REAL,REAL 1 &
 (proj(1,1)) " is one-to-one &
 proj1 ((proj(1,1)) ") = REAL &
 proj2 ((proj(1,1)) ") = REAL 1 &
 (ex b1 being Function-like quasi_total Relation of REAL,REAL 1 st
    b1 is bijective(REAL, REAL 1) & (proj(1,1)) " = b1);

:: PDIFF_1:funcreg 1
registration
  cluster proj(1,1) -> Function-like quasi_total bijective;
end;

:: PDIFF_1:funcnot 2 => PDIFF_1:func 2
definition
  let a1 be Function-like Relation of REAL,REAL;
  func <>* A1 -> Function-like Relation of REAL 1,REAL 1 equals
    (proj(1,1)) * (a1 * ((proj(1,1)) "));
end;

:: PDIFF_1:def 2
theorem
for b1 being Function-like Relation of REAL,REAL holds
   <>* b1 = (proj(1,1)) * (b1 * ((proj(1,1)) "));

:: PDIFF_1:funcnot 3 => PDIFF_1:func 3
definition
  let a1 be Element of NAT;
  let a2 be Function-like Relation of REAL a1,REAL;
  func <>* A2 -> Function-like Relation of REAL a1,REAL 1 equals
    a2 * ((proj(1,1)) ");
end;

:: PDIFF_1:def 3
theorem
for b1 being Element of NAT
for b2 being Function-like Relation of REAL b1,REAL holds
   <>* b2 = b2 * ((proj(1,1)) ");

:: PDIFF_1:funcnot 4 => PDIFF_1:func 4
definition
  let a1, a2 be Element of NAT;
  func Proj(A1,A2) -> Function-like quasi_total Relation of the carrier of REAL-NS a2,the carrier of REAL-NS 1 means
    for b1 being Element of the carrier of REAL-NS a2 holds
       it . b1 = <*(proj(a1,a2)) . b1*>;
end;

:: PDIFF_1:def 4
theorem
for b1, b2 being Element of NAT
for b3 being Function-like quasi_total Relation of the carrier of REAL-NS b2,the carrier of REAL-NS 1 holds
      b3 = Proj(b1,b2)
   iff
      for b4 being Element of the carrier of REAL-NS b2 holds
         b3 . b4 = <*(proj(b1,b2)) . b4*>;

:: PDIFF_1:funcnot 5 => PDIFF_1:func 5
definition
  let a1 be Element of NAT;
  let a2 be FinSequence of REAL;
  func reproj(A1,A2) -> Relation-like Function-like set means
    proj1 it = REAL &
     (for b1 being Element of REAL holds
        it . b1 = Replace(a2,a1,b1));
end;

:: PDIFF_1:def 5
theorem
for b1 being Element of NAT
for b2 being FinSequence of REAL
for b3 being Relation-like Function-like set holds
      b3 = reproj(b1,b2)
   iff
      proj1 b3 = REAL &
       (for b4 being Element of REAL holds
          b3 . b4 = Replace(b2,b1,b4));

:: PDIFF_1:funcnot 6 => PDIFF_1:func 6
definition
  let a1, a2 be Element of NAT;
  let a3 be Element of REAL a1;
  redefine func reproj(a2,a3) -> Function-like quasi_total Relation of REAL,REAL a1;
end;

:: PDIFF_1:funcnot 7 => PDIFF_1:func 7
definition
  let a1, a2 be Element of NAT;
  let a3 be Element of the carrier of REAL-NS a1;
  func reproj(A2,A3) -> Function-like quasi_total Relation of the carrier of REAL-NS 1,the carrier of REAL-NS a1 means
    for b1 being Element of the carrier of REAL-NS 1 holds
       ex b2 being Element of REAL st
          ex b3 being Element of REAL a1 st
             b1 = <*b2*> & b3 = a3 & it . b1 = (reproj(a2,b3)) . b2;
end;

:: PDIFF_1:def 6
theorem
for b1, b2 being Element of NAT
for b3 being Element of the carrier of REAL-NS b1
for b4 being Function-like quasi_total Relation of the carrier of REAL-NS 1,the carrier of REAL-NS b1 holds
      b4 = reproj(b2,b3)
   iff
      for b5 being Element of the carrier of REAL-NS 1 holds
         ex b6 being Element of REAL st
            ex b7 being Element of REAL b1 st
               b5 = <*b6*> & b7 = b3 & b4 . b5 = (reproj(b2,b7)) . b6;

:: PDIFF_1:prednot 1 => PDIFF_1:pred 1
definition
  let a1, a2 be non empty Element of NAT;
  let a3 be Function-like Relation of REAL a1,REAL a2;
  let a4 be Element of REAL a1;
  pred A3 is_differentiable_in A4 means
    ex b1 being Function-like Relation of the carrier of REAL-NS a1,the carrier of REAL-NS a2 st
       ex b2 being Element of the carrier of REAL-NS a1 st
          a3 = b1 & a4 = b2 & b1 is_differentiable_in b2;
end;

:: PDIFF_1:dfs 7
definiens
  let a1, a2 be non empty Element of NAT;
  let a3 be Function-like Relation of REAL a1,REAL a2;
  let a4 be Element of REAL a1;
To prove
     a3 is_differentiable_in a4
it is sufficient to prove
  thus ex b1 being Function-like Relation of the carrier of REAL-NS a1,the carrier of REAL-NS a2 st
       ex b2 being Element of the carrier of REAL-NS a1 st
          a3 = b1 & a4 = b2 & b1 is_differentiable_in b2;

:: PDIFF_1:def 7
theorem
for b1, b2 being non empty Element of NAT
for b3 being Function-like Relation of REAL b1,REAL b2
for b4 being Element of REAL b1 holds
      b3 is_differentiable_in b4
   iff
      ex b5 being Function-like Relation of the carrier of REAL-NS b1,the carrier of REAL-NS b2 st
         ex b6 being Element of the carrier of REAL-NS b1 st
            b3 = b5 & b4 = b6 & b5 is_differentiable_in b6;

:: PDIFF_1:funcnot 8 => PDIFF_1:func 8
definition
  let a1, a2 be non empty Element of NAT;
  let a3 be Function-like Relation of REAL a1,REAL a2;
  let a4 be Element of REAL a1;
  assume a3 is_differentiable_in a4;
  func diff(A3,A4) -> Function-like quasi_total Relation of REAL a1,REAL a2 means
    ex b1 being Function-like Relation of the carrier of REAL-NS a1,the carrier of REAL-NS a2 st
       ex b2 being Element of the carrier of REAL-NS a1 st
          a3 = b1 & a4 = b2 & it = diff(b1,b2);
end;

:: PDIFF_1:def 8
theorem
for b1, b2 being non empty Element of NAT
for b3 being Function-like Relation of REAL b1,REAL b2
for b4 being Element of REAL b1
   st b3 is_differentiable_in b4
for b5 being Function-like quasi_total Relation of REAL b1,REAL b2 holds
      b5 = diff(b3,b4)
   iff
      ex b6 being Function-like Relation of the carrier of REAL-NS b1,the carrier of REAL-NS b2 st
         ex b7 being Element of the carrier of REAL-NS b1 st
            b3 = b6 & b4 = b7 & b5 = diff(b6,b7);

:: PDIFF_1:th 3
theorem
for b1 being Function-like quasi_total Relation of REAL,REAL 1
      st b1 = (proj(1,1)) "
   holds (for b2 being Element of the carrier of REAL-NS 1
    for b3 being Element of REAL
          st b2 = b1 . b3
       holds ||.b2.|| = abs b3) &
    (for b2, b3 being Element of the carrier of REAL-NS 1
    for b4, b5 being Element of REAL
          st b2 = b1 . b4 & b3 = b1 . b5
       holds b2 + b3 = b1 . (b4 + b5)) &
    (for b2 being Element of the carrier of REAL-NS 1
    for b3, b4 being Element of REAL
          st b2 = b1 . b3
       holds b4 * b2 = b1 . (b4 * b3)) &
    (for b2 being Element of the carrier of REAL-NS 1
    for b3 being Element of REAL
          st b2 = b1 . b3
       holds - b2 = b1 . - b3) &
    (for b2, b3 being Element of the carrier of REAL-NS 1
    for b4, b5 being Element of REAL
          st b2 = b1 . b4 & b3 = b1 . b5
       holds b2 - b3 = b1 . (b4 - b5));

:: PDIFF_1:th 4
theorem
for b1 being Function-like quasi_total Relation of REAL 1,REAL
      st b1 = proj(1,1)
   holds (for b2 being Element of the carrier of REAL-NS 1
    for b3 being Element of REAL
          st b1 . b2 = b3
       holds ||.b2.|| = abs b3) &
    (for b2, b3 being Element of the carrier of REAL-NS 1
    for b4, b5 being Element of REAL
          st b1 . b2 = b4 & b1 . b3 = b5
       holds b1 . (b2 + b3) = b4 + b5) &
    (for b2 being Element of the carrier of REAL-NS 1
    for b3, b4 being Element of REAL
          st b1 . b2 = b3
       holds b1 . (b4 * b2) = b4 * b3) &
    (for b2 being Element of the carrier of REAL-NS 1
    for b3 being Element of REAL
          st b1 . b2 = b3
       holds b1 . - b2 = - b3) &
    (for b2, b3 being Element of the carrier of REAL-NS 1
    for b4, b5 being Element of REAL
          st b1 . b2 = b4 & b1 . b3 = b5
       holds b1 . (b2 - b3) = b4 - b5);

:: PDIFF_1:th 5
theorem
for b1 being Function-like quasi_total Relation of REAL,REAL 1
for b2 being Function-like quasi_total Relation of REAL 1,REAL
      st b1 = (proj(1,1)) " & b2 = proj(1,1)
   holds (for b3 being Function-like REST-like Relation of the carrier of REAL-NS 1,the carrier of REAL-NS 1 holds
       (b2 * b3) * b1 is Function-like REST-like Relation of REAL,REAL) &
    (for b3 being Function-like quasi_total additive homogeneous Relation of the carrier of REAL-NS 1,the carrier of REAL-NS 1 holds
       (b2 * b3) * b1 is Function-like linear Relation of REAL,REAL);

:: PDIFF_1:th 6
theorem
for b1 being Function-like quasi_total Relation of REAL,REAL 1
for b2 being Function-like quasi_total Relation of REAL 1,REAL
      st b1 = (proj(1,1)) " & b2 = proj(1,1)
   holds (for b3 being Function-like REST-like Relation of REAL,REAL holds
       (b1 * b3) * b2 is Function-like REST-like Relation of the carrier of REAL-NS 1,the carrier of REAL-NS 1) &
    (for b3 being Function-like linear Relation of REAL,REAL holds
       (b1 * b3) * b2 is Function-like quasi_total additive homogeneous bounded Relation of the carrier of REAL-NS 1,the carrier of REAL-NS 1);

:: PDIFF_1:th 7
theorem
for b1 being Function-like Relation of the carrier of REAL-NS 1,the carrier of REAL-NS 1
for b2 being Function-like Relation of REAL,REAL
for b3 being Element of the carrier of REAL-NS 1
for b4 being Element of REAL
      st b1 = <>* b2 & b3 = <*b4*> & b1 is_differentiable_in b3
   holds b2 is_differentiable_in b4 &
    diff(b2,b4) = ((proj(1,1)) " * ((diff(b1,b3)) * proj(1,1))) . 1;

:: PDIFF_1:th 8
theorem
for b1 being Function-like Relation of the carrier of REAL-NS 1,the carrier of REAL-NS 1
for b2 being Function-like Relation of REAL,REAL
for b3 being Element of the carrier of REAL-NS 1
for b4 being Element of REAL
      st b1 = <>* b2 & b3 = <*b4*> & b2 is_differentiable_in b4
   holds b1 is_differentiable_in b3 &
    (diff(b1,b3)) . <*1*> = <*diff(b2,b4)*>;

:: PDIFF_1:th 9
theorem
for b1 being Function-like Relation of the carrier of REAL-NS 1,the carrier of REAL-NS 1
for b2 being Function-like Relation of REAL,REAL
for b3 being Element of the carrier of REAL-NS 1
for b4 being Element of REAL
      st b1 = <>* b2 & b3 = <*b4*>
   holds    b1 is_differentiable_in b3
   iff
      b2 is_differentiable_in b4;

:: PDIFF_1:th 10
theorem
for b1 being Function-like Relation of the carrier of REAL-NS 1,the carrier of REAL-NS 1
for b2 being Function-like Relation of REAL,REAL
for b3 being Element of the carrier of REAL-NS 1
for b4 being Element of REAL
      st b1 = <>* b2 & b3 = <*b4*> & b1 is_differentiable_in b3
   holds (diff(b1,b3)) . <*1*> = <*diff(b2,b4)*>;

:: PDIFF_1:prednot 2 => PDIFF_1:pred 2
definition
  let a1, a2 be non empty Element of NAT;
  let a3 be Element of NAT;
  let a4 be Function-like Relation of the carrier of REAL-NS a2,the carrier of REAL-NS a1;
  let a5 be Element of the carrier of REAL-NS a2;
  pred A4 is_partial_differentiable_in A5,A3 means
    a4 * reproj(a3,a5) is_differentiable_in (Proj(a3,a2)) . a5;
end;

:: PDIFF_1:dfs 9
definiens
  let a1, a2 be non empty Element of NAT;
  let a3 be Element of NAT;
  let a4 be Function-like Relation of the carrier of REAL-NS a2,the carrier of REAL-NS a1;
  let a5 be Element of the carrier of REAL-NS a2;
To prove
     a4 is_partial_differentiable_in a5,a3
it is sufficient to prove
  thus a4 * reproj(a3,a5) is_differentiable_in (Proj(a3,a2)) . a5;

:: PDIFF_1:def 9
theorem
for b1, b2 being non empty Element of NAT
for b3 being Element of NAT
for b4 being Function-like Relation of the carrier of REAL-NS b2,the carrier of REAL-NS b1
for b5 being Element of the carrier of REAL-NS b2 holds
      b4 is_partial_differentiable_in b5,b3
   iff
      b4 * reproj(b3,b5) is_differentiable_in (Proj(b3,b2)) . b5;

:: PDIFF_1:funcnot 9 => PDIFF_1:func 9
definition
  let a1, a2 be non empty Element of NAT;
  let a3 be Element of NAT;
  let a4 be Function-like Relation of the carrier of REAL-NS a1,the carrier of REAL-NS a2;
  let a5 be Element of the carrier of REAL-NS a1;
  func partdiff(A4,A5,A3) -> Element of the carrier of R_NormSpace_of_BoundedLinearOperators(REAL-NS 1,REAL-NS a2) equals
    diff(a4 * reproj(a3,a5),(Proj(a3,a1)) . a5);
end;

:: PDIFF_1:def 10
theorem
for b1, b2 being non empty Element of NAT
for b3 being Element of NAT
for b4 being Function-like Relation of the carrier of REAL-NS b1,the carrier of REAL-NS b2
for b5 being Element of the carrier of REAL-NS b1 holds
   partdiff(b4,b5,b3) = diff(b4 * reproj(b3,b5),(Proj(b3,b1)) . b5);

:: PDIFF_1:prednot 3 => PDIFF_1:pred 3
definition
  let a1 be non empty Element of NAT;
  let a2 be Element of NAT;
  let a3 be Function-like Relation of REAL a1,REAL;
  let a4 be Element of REAL a1;
  pred A3 is_partial_differentiable_in A4,A2 means
    a3 * reproj(a2,a4) is_differentiable_in (proj(a2,a1)) . a4;
end;

:: PDIFF_1:dfs 11
definiens
  let a1 be non empty Element of NAT;
  let a2 be Element of NAT;
  let a3 be Function-like Relation of REAL a1,REAL;
  let a4 be Element of REAL a1;
To prove
     a3 is_partial_differentiable_in a4,a2
it is sufficient to prove
  thus a3 * reproj(a2,a4) is_differentiable_in (proj(a2,a1)) . a4;

:: PDIFF_1:def 11
theorem
for b1 being non empty Element of NAT
for b2 being Element of NAT
for b3 being Function-like Relation of REAL b1,REAL
for b4 being Element of REAL b1 holds
      b3 is_partial_differentiable_in b4,b2
   iff
      b3 * reproj(b2,b4) is_differentiable_in (proj(b2,b1)) . b4;

:: PDIFF_1:funcnot 10 => PDIFF_1:func 10
definition
  let a1 be non empty Element of NAT;
  let a2 be Element of NAT;
  let a3 be Function-like Relation of REAL a1,REAL;
  let a4 be Element of REAL a1;
  func partdiff(A3,A4,A2) -> real set equals
    diff(a3 * reproj(a2,a4),(proj(a2,a1)) . a4);
end;

:: PDIFF_1:def 12
theorem
for b1 being non empty Element of NAT
for b2 being Element of NAT
for b3 being Function-like Relation of REAL b1,REAL
for b4 being Element of REAL b1 holds
   partdiff(b3,b4,b2) = diff(b3 * reproj(b2,b4),(proj(b2,b1)) . b4);

:: PDIFF_1:th 11
theorem
for b1 being non empty Element of NAT
for b2 being Element of NAT holds
   Proj(b2,b1) = (proj(b2,b1)) * ((proj(1,1)) ");

:: PDIFF_1:th 12
theorem
for b1 being non empty Element of NAT
for b2 being Element of NAT
for b3 being Element of the carrier of REAL-NS b1
for b4 being Element of REAL b1
      st b3 = b4
   holds (reproj(b2,b4)) * proj(1,1) = reproj(b2,b3);

:: PDIFF_1:th 13
theorem
for b1 being non empty Element of NAT
for b2 being Element of NAT
for b3 being Function-like Relation of the carrier of REAL-NS b1,the carrier of REAL-NS 1
for b4 being Function-like Relation of REAL b1,REAL
for b5 being Element of the carrier of REAL-NS b1
for b6 being Element of REAL b1
      st b3 = <>* b4 & b5 = b6
   holds <>* (b4 * reproj(b2,b6)) = b3 * reproj(b2,b5);

:: PDIFF_1:th 14
theorem
for b1 being non empty Element of NAT
for b2 being Element of NAT
for b3 being Function-like Relation of the carrier of REAL-NS b1,the carrier of REAL-NS 1
for b4 being Function-like Relation of REAL b1,REAL
for b5 being Element of the carrier of REAL-NS b1
for b6 being Element of REAL b1
      st b3 = <>* b4 & b5 = b6
   holds    b3 is_partial_differentiable_in b5,b2
   iff
      b4 is_partial_differentiable_in b6,b2;

:: PDIFF_1:th 15
theorem
for b1 being non empty Element of NAT
for b2 being Element of NAT
for b3 being Function-like Relation of the carrier of REAL-NS b1,the carrier of REAL-NS 1
for b4 being Function-like Relation of REAL b1,REAL
for b5 being Element of the carrier of REAL-NS b1
for b6 being Element of REAL b1
      st b3 = <>* b4 & b5 = b6 & b3 is_partial_differentiable_in b5,b2
   holds (partdiff(b3,b5,b2)) . <*1*> = <*partdiff(b4,b6,b2)*>;

:: PDIFF_1:prednot 4 => PDIFF_1:pred 4
definition
  let a1, a2 be non empty Element of NAT;
  let a3 be Element of NAT;
  let a4 be Function-like Relation of REAL a1,REAL a2;
  let a5 be Element of REAL a1;
  pred A4 is_partial_differentiable_in A5,A3 means
    ex b1 being Function-like Relation of the carrier of REAL-NS a1,the carrier of REAL-NS a2 st
       ex b2 being Element of the carrier of REAL-NS a1 st
          a4 = b1 & a5 = b2 & b1 is_partial_differentiable_in b2,a3;
end;

:: PDIFF_1:dfs 13
definiens
  let a1, a2 be non empty Element of NAT;
  let a3 be Element of NAT;
  let a4 be Function-like Relation of REAL a1,REAL a2;
  let a5 be Element of REAL a1;
To prove
     a4 is_partial_differentiable_in a5,a3
it is sufficient to prove
  thus ex b1 being Function-like Relation of the carrier of REAL-NS a1,the carrier of REAL-NS a2 st
       ex b2 being Element of the carrier of REAL-NS a1 st
          a4 = b1 & a5 = b2 & b1 is_partial_differentiable_in b2,a3;

:: PDIFF_1:def 13
theorem
for b1, b2 being non empty Element of NAT
for b3 being Element of NAT
for b4 being Function-like Relation of REAL b1,REAL b2
for b5 being Element of REAL b1 holds
      b4 is_partial_differentiable_in b5,b3
   iff
      ex b6 being Function-like Relation of the carrier of REAL-NS b1,the carrier of REAL-NS b2 st
         ex b7 being Element of the carrier of REAL-NS b1 st
            b4 = b6 & b5 = b7 & b6 is_partial_differentiable_in b7,b3;

:: PDIFF_1:funcnot 11 => PDIFF_1:func 11
definition
  let a1, a2 be non empty Element of NAT;
  let a3 be Element of NAT;
  let a4 be Function-like Relation of REAL a1,REAL a2;
  let a5 be Element of REAL a1;
  assume a4 is_partial_differentiable_in a5,a3;
  func partdiff(A4,A5,A3) -> Element of REAL a2 means
    ex b1 being Function-like Relation of the carrier of REAL-NS a1,the carrier of REAL-NS a2 st
       ex b2 being Element of the carrier of REAL-NS a1 st
          a4 = b1 &
           a5 = b2 &
           it = (partdiff(b1,b2,a3)) . <*1*>;
end;

:: PDIFF_1:def 14
theorem
for b1, b2 being non empty Element of NAT
for b3 being Element of NAT
for b4 being Function-like Relation of REAL b1,REAL b2
for b5 being Element of REAL b1
   st b4 is_partial_differentiable_in b5,b3
for b6 being Element of REAL b2 holds
      b6 = partdiff(b4,b5,b3)
   iff
      ex b7 being Function-like Relation of the carrier of REAL-NS b1,the carrier of REAL-NS b2 st
         ex b8 being Element of the carrier of REAL-NS b1 st
            b4 = b7 &
             b5 = b8 &
             b6 = (partdiff(b7,b8,b3)) . <*1*>;

:: PDIFF_1:th 16
theorem
for b1 being Element of NAT
for b2, b3 being non empty Element of NAT
for b4 being Function-like Relation of the carrier of REAL-NS b2,the carrier of REAL-NS b3
for b5 being Function-like Relation of REAL b2,REAL b3
for b6 being Element of the carrier of REAL-NS b2
for b7 being Element of REAL b2
      st b4 = b5 & b6 = b7
   holds    b4 is_partial_differentiable_in b6,b1
   iff
      b5 is_partial_differentiable_in b7,b1;

:: PDIFF_1:th 17
theorem
for b1 being Element of NAT
for b2, b3 being non empty Element of NAT
for b4 being Function-like Relation of the carrier of REAL-NS b2,the carrier of REAL-NS b3
for b5 being Function-like Relation of REAL b2,REAL b3
for b6 being Element of the carrier of REAL-NS b2
for b7 being Element of REAL b2
      st b4 = b5 & b6 = b7 & b4 is_partial_differentiable_in b6,b1
   holds (partdiff(b4,b6,b1)) . <*1*> = partdiff(b5,b7,b1);

:: PDIFF_1:th 18
theorem
for b1 being non empty Element of NAT
for b2 being Element of NAT
for b3 being Function-like Relation of REAL b1,REAL
for b4 being Element of REAL b1
for b5 being Function-like Relation of REAL b1,REAL 1
      st b5 = <>* b3
   holds    b5 is_partial_differentiable_in b4,b2
   iff
      b3 is_partial_differentiable_in b4,b2;

:: PDIFF_1:th 19
theorem
for b1 being non empty Element of NAT
for b2 being Element of NAT
for b3 being Function-like Relation of REAL b1,REAL
for b4 being Element of REAL b1
for b5 being Function-like Relation of REAL b1,REAL 1
      st b5 = <>* b3 & b5 is_partial_differentiable_in b4,b2
   holds partdiff(b5,b4,b2) = <*partdiff(b3,b4,b2)*>;

:: PDIFF_1:prednot 5 => PDIFF_1:pred 5
definition
  let a1, a2 be non empty Element of NAT;
  let a3, a4 be Element of NAT;
  let a5 be Function-like Relation of the carrier of REAL-NS a1,the carrier of REAL-NS a2;
  let a6 be Element of the carrier of REAL-NS a1;
  pred A5 is_partial_differentiable_in A6,A3,A4 means
    ((Proj(a4,a2)) * a5) * reproj(a3,a6) is_differentiable_in (Proj(a3,a1)) . a6;
end;

:: PDIFF_1:dfs 15
definiens
  let a1, a2 be non empty Element of NAT;
  let a3, a4 be Element of NAT;
  let a5 be Function-like Relation of the carrier of REAL-NS a1,the carrier of REAL-NS a2;
  let a6 be Element of the carrier of REAL-NS a1;
To prove
     a5 is_partial_differentiable_in a6,a3,a4
it is sufficient to prove
  thus ((Proj(a4,a2)) * a5) * reproj(a3,a6) is_differentiable_in (Proj(a3,a1)) . a6;

:: PDIFF_1:def 15
theorem
for b1, b2 being non empty Element of NAT
for b3, b4 being Element of NAT
for b5 being Function-like Relation of the carrier of REAL-NS b1,the carrier of REAL-NS b2
for b6 being Element of the carrier of REAL-NS b1 holds
      b5 is_partial_differentiable_in b6,b3,b4
   iff
      ((Proj(b4,b2)) * b5) * reproj(b3,b6) is_differentiable_in (Proj(b3,b1)) . b6;

:: PDIFF_1:funcnot 12 => PDIFF_1:func 12
definition
  let a1, a2 be non empty Element of NAT;
  let a3, a4 be Element of NAT;
  let a5 be Function-like Relation of the carrier of REAL-NS a1,the carrier of REAL-NS a2;
  let a6 be Element of the carrier of REAL-NS a1;
  func partdiff(A5,A6,A3,A4) -> Element of the carrier of R_NormSpace_of_BoundedLinearOperators(REAL-NS 1,REAL-NS 1) equals
    diff(((Proj(a4,a2)) * a5) * reproj(a3,a6),(Proj(a3,a1)) . a6);
end;

:: PDIFF_1:def 16
theorem
for b1, b2 being non empty Element of NAT
for b3, b4 being Element of NAT
for b5 being Function-like Relation of the carrier of REAL-NS b1,the carrier of REAL-NS b2
for b6 being Element of the carrier of REAL-NS b1 holds
   partdiff(b5,b6,b3,b4) = diff(((Proj(b4,b2)) * b5) * reproj(b3,b6),(Proj(b3,b1)) . b6);

:: PDIFF_1:prednot 6 => PDIFF_1:pred 6
definition
  let a1, a2 be non empty Element of NAT;
  let a3, a4 be Element of NAT;
  let a5 be Function-like Relation of REAL a1,REAL a2;
  let a6 be Element of REAL a1;
  pred A5 is_partial_differentiable_in A6,A3,A4 means
    ((proj(a4,a2)) * a5) * reproj(a3,a6) is_differentiable_in (proj(a3,a1)) . a6;
end;

:: PDIFF_1:dfs 17
definiens
  let a1, a2 be non empty Element of NAT;
  let a3, a4 be Element of NAT;
  let a5 be Function-like Relation of REAL a1,REAL a2;
  let a6 be Element of REAL a1;
To prove
     a5 is_partial_differentiable_in a6,a3,a4
it is sufficient to prove
  thus ((proj(a4,a2)) * a5) * reproj(a3,a6) is_differentiable_in (proj(a3,a1)) . a6;

:: PDIFF_1:def 17
theorem
for b1, b2 being non empty Element of NAT
for b3, b4 being Element of NAT
for b5 being Function-like Relation of REAL b1,REAL b2
for b6 being Element of REAL b1 holds
      b5 is_partial_differentiable_in b6,b3,b4
   iff
      ((proj(b4,b2)) * b5) * reproj(b3,b6) is_differentiable_in (proj(b3,b1)) . b6;

:: PDIFF_1:funcnot 13 => PDIFF_1:func 13
definition
  let a1, a2 be non empty Element of NAT;
  let a3, a4 be Element of NAT;
  let a5 be Function-like Relation of REAL a1,REAL a2;
  let a6 be Element of REAL a1;
  func partdiff(A5,A6,A3,A4) -> real set equals
    diff(((proj(a4,a2)) * a5) * reproj(a3,a6),(proj(a3,a1)) . a6);
end;

:: PDIFF_1:def 18
theorem
for b1, b2 being non empty Element of NAT
for b3, b4 being Element of NAT
for b5 being Function-like Relation of REAL b1,REAL b2
for b6 being Element of REAL b1 holds
   partdiff(b5,b6,b3,b4) = diff(((proj(b4,b2)) * b5) * reproj(b3,b6),(proj(b3,b1)) . b6);

:: PDIFF_1:th 20
theorem
for b1, b2 being non empty Element of NAT
for b3 being Function-like Relation of the carrier of REAL-NS b1,the carrier of REAL-NS b2
for b4 being Function-like Relation of REAL b1,REAL b2
for b5 being Element of the carrier of REAL-NS b1
for b6 being Element of REAL b1
      st b3 = b4 & b5 = b6
   holds    b3 is_differentiable_in b5
   iff
      b4 is_differentiable_in b6;

:: PDIFF_1:th 21
theorem
for b1, b2 being non empty Element of NAT
for b3 being Function-like Relation of the carrier of REAL-NS b1,the carrier of REAL-NS b2
for b4 being Function-like Relation of REAL b1,REAL b2
for b5 being Element of the carrier of REAL-NS b1
for b6 being Element of REAL b1
      st b3 = b4 & b5 = b6 & b3 is_differentiable_in b5
   holds diff(b3,b5) = diff(b4,b6);

:: PDIFF_1:th 22
theorem
for b1, b2 being non empty Element of NAT
for b3, b4 being Element of NAT
for b5 being Function-like Relation of the carrier of REAL-NS b1,the carrier of REAL-NS b2
for b6 being Function-like Relation of REAL b1,REAL b2
for b7 being Element of the carrier of REAL-NS b1
for b8 being Element of REAL b1
      st b5 = b6 & b7 = b8
   holds ((Proj(b3,b2)) * b5) * reproj(b4,b7) = <>* (((proj(b3,b2)) * b6) * reproj(b4,b8));

:: PDIFF_1:th 23
theorem
for b1, b2 being non empty Element of NAT
for b3, b4 being Element of NAT
for b5 being Function-like Relation of the carrier of REAL-NS b2,the carrier of REAL-NS b1
for b6 being Function-like Relation of REAL b2,REAL b1
for b7 being Element of the carrier of REAL-NS b2
for b8 being Element of REAL b2
      st b5 = b6 & b7 = b8
   holds    b5 is_partial_differentiable_in b7,b3,b4
   iff
      b6 is_partial_differentiable_in b8,b3,b4;

:: PDIFF_1:th 24
theorem
for b1, b2 being non empty Element of NAT
for b3, b4 being Element of NAT
for b5 being Function-like Relation of the carrier of REAL-NS b2,the carrier of REAL-NS b1
for b6 being Function-like Relation of REAL b2,REAL b1
for b7 being Element of the carrier of REAL-NS b2
for b8 being Element of REAL b2
      st b5 = b6 & b7 = b8 & b5 is_partial_differentiable_in b7,b3,b4
   holds (partdiff(b5,b7,b3,b4)) . <*1*> = <*partdiff(b6,b8,b3,b4)*>;

:: PDIFF_1:prednot 7 => PDIFF_1:pred 7
definition
  let a1, a2 be non empty Element of NAT;
  let a3 be Element of NAT;
  let a4 be Function-like Relation of the carrier of REAL-NS a1,the carrier of REAL-NS a2;
  let a5 be set;
  pred A4 is_partial_differentiable_on A5,A3 means
    a5 c= dom a4 &
     (for b1 being Element of the carrier of REAL-NS a1
           st b1 in a5
        holds a4 | a5 is_partial_differentiable_in b1,a3);
end;

:: PDIFF_1:dfs 19
definiens
  let a1, a2 be non empty Element of NAT;
  let a3 be Element of NAT;
  let a4 be Function-like Relation of the carrier of REAL-NS a1,the carrier of REAL-NS a2;
  let a5 be set;
To prove
     a4 is_partial_differentiable_on a5,a3
it is sufficient to prove
  thus a5 c= dom a4 &
     (for b1 being Element of the carrier of REAL-NS a1
           st b1 in a5
        holds a4 | a5 is_partial_differentiable_in b1,a3);

:: PDIFF_1:def 19
theorem
for b1, b2 being non empty Element of NAT
for b3 being Element of NAT
for b4 being Function-like Relation of the carrier of REAL-NS b1,the carrier of REAL-NS b2
for b5 being set holds
      b4 is_partial_differentiable_on b5,b3
   iff
      b5 c= dom b4 &
       (for b6 being Element of the carrier of REAL-NS b1
             st b6 in b5
          holds b4 | b5 is_partial_differentiable_in b6,b3);

:: PDIFF_1:th 25
theorem
for b1, b2 being non empty Element of NAT
for b3 being Element of NAT
for b4 being set
for b5 being Function-like Relation of the carrier of REAL-NS b2,the carrier of REAL-NS b1
      st b5 is_partial_differentiable_on b4,b3
   holds b4 is Element of bool the carrier of REAL-NS b2;

:: PDIFF_1:funcnot 14 => PDIFF_1:func 14
definition
  let a1, a2 be non empty Element of NAT;
  let a3 be Element of NAT;
  let a4 be Function-like Relation of the carrier of REAL-NS a1,the carrier of REAL-NS a2;
  let a5 be set;
  assume a4 is_partial_differentiable_on a5,a3;
  func A4 `partial|(A5,A3) -> Function-like Relation of the carrier of REAL-NS a1,the carrier of R_NormSpace_of_BoundedLinearOperators(REAL-NS 1,REAL-NS a2) means
    dom it = a5 &
     (for b1 being Element of the carrier of REAL-NS a1
           st b1 in a5
        holds it /. b1 = partdiff(a4,b1,a3));
end;

:: PDIFF_1:def 20
theorem
for b1, b2 being non empty Element of NAT
for b3 being Element of NAT
for b4 being Function-like Relation of the carrier of REAL-NS b1,the carrier of REAL-NS b2
for b5 being set
   st b4 is_partial_differentiable_on b5,b3
for b6 being Function-like Relation of the carrier of REAL-NS b1,the carrier of R_NormSpace_of_BoundedLinearOperators(REAL-NS 1,REAL-NS b2) holds
      b6 = b4 `partial|(b5,b3)
   iff
      dom b6 = b5 &
       (for b7 being Element of the carrier of REAL-NS b1
             st b7 in b5
          holds b6 /. b7 = partdiff(b4,b7,b3));

:: PDIFF_1:th 26
theorem
for b1, b2 being non empty Element of NAT
for b3 being Element of NAT
for b4, b5 being Function-like Relation of the carrier of REAL-NS b2,the carrier of REAL-NS b1
for b6 being Element of the carrier of REAL-NS b2 holds
   (b4 + b5) * reproj(b3,b6) = (b4 * reproj(b3,b6)) + (b5 * reproj(b3,b6)) &
    (b4 - b5) * reproj(b3,b6) = (b4 * reproj(b3,b6)) - (b5 * reproj(b3,b6));

:: PDIFF_1:th 27
theorem
for b1, b2 being non empty Element of NAT
for b3 being Element of NAT
for b4 being Element of REAL
for b5 being Function-like Relation of the carrier of REAL-NS b2,the carrier of REAL-NS b1
for b6 being Element of the carrier of REAL-NS b2 holds
   b4 (#) (b5 * reproj(b3,b6)) = (b4 (#) b5) * reproj(b3,b6);

:: PDIFF_1:th 28
theorem
for b1, b2 being non empty Element of NAT
for b3 being Element of NAT
for b4, b5 being Function-like Relation of the carrier of REAL-NS b2,the carrier of REAL-NS b1
for b6 being Element of the carrier of REAL-NS b2
      st b4 is_partial_differentiable_in b6,b3 & b5 is_partial_differentiable_in b6,b3
   holds b4 + b5 is_partial_differentiable_in b6,b3 &
    partdiff(b4 + b5,b6,b3) = (partdiff(b4,b6,b3)) + partdiff(b5,b6,b3);

:: PDIFF_1:th 29
theorem
for b1 being non empty Element of NAT
for b2 being Element of NAT
for b3, b4 being Function-like Relation of REAL b1,REAL
for b5 being Element of REAL b1
      st b3 is_partial_differentiable_in b5,b2 & b4 is_partial_differentiable_in b5,b2
   holds b3 + b4 is_partial_differentiable_in b5,b2 &
    partdiff(b3 + b4,b5,b2) = (partdiff(b3,b5,b2)) + partdiff(b4,b5,b2);

:: PDIFF_1:th 30
theorem
for b1, b2 being non empty Element of NAT
for b3 being Element of NAT
for b4, b5 being Function-like Relation of the carrier of REAL-NS b2,the carrier of REAL-NS b1
for b6 being Element of the carrier of REAL-NS b2
      st b4 is_partial_differentiable_in b6,b3 & b5 is_partial_differentiable_in b6,b3
   holds b4 - b5 is_partial_differentiable_in b6,b3 &
    partdiff(b4 - b5,b6,b3) = (partdiff(b4,b6,b3)) - partdiff(b5,b6,b3);

:: PDIFF_1:th 31
theorem
for b1 being non empty Element of NAT
for b2 being Element of NAT
for b3, b4 being Function-like Relation of REAL b1,REAL
for b5 being Element of REAL b1
      st b3 is_partial_differentiable_in b5,b2 & b4 is_partial_differentiable_in b5,b2
   holds b3 - b4 is_partial_differentiable_in b5,b2 &
    partdiff(b3 - b4,b5,b2) = (partdiff(b3,b5,b2)) - partdiff(b4,b5,b2);

:: PDIFF_1:th 32
theorem
for b1, b2 being non empty Element of NAT
for b3 being Element of NAT
for b4 being Element of REAL
for b5 being Function-like Relation of the carrier of REAL-NS b2,the carrier of REAL-NS b1
for b6 being Element of the carrier of REAL-NS b2
      st b5 is_partial_differentiable_in b6,b3
   holds b4 (#) b5 is_partial_differentiable_in b6,b3 &
    partdiff(b4 (#) b5,b6,b3) = b4 * partdiff(b5,b6,b3);

:: PDIFF_1:th 33
theorem
for b1 being non empty Element of NAT
for b2 being Element of NAT
for b3 being Element of REAL
for b4 being Function-like Relation of REAL b1,REAL
for b5 being Element of REAL b1
      st b4 is_partial_differentiable_in b5,b2
   holds b3 (#) b4 is_partial_differentiable_in b5,b2 &
    partdiff(b3 (#) b4,b5,b2) = b3 * partdiff(b4,b5,b2);