reserve f for PartFunc of REAL-NS 1,REAL-NS 1;
reserve g for PartFunc of REAL,REAL;
reserve x for Point of REAL-NS 1;
reserve y for Real;
reserve m,n for non zero Nat;
reserve i,j for Nat;
reserve f for PartFunc of REAL-NS n,REAL-NS 1;
reserve g for PartFunc of REAL n,REAL;
reserve x for Point of REAL-NS n;
reserve y for Element of REAL n;
reserve X for set;
reserve r for Real;
reserve f,f1,f2 for PartFunc of REAL-NS m,REAL-NS n;
reserve g,g1,g2 for PartFunc of REAL n,REAL;
reserve h for PartFunc of REAL m,REAL n;
reserve x for Point of REAL-NS m;
reserve y for Element of REAL n;
reserve z for Element of REAL m;

theorem
  for m,n be non zero Nat, F be PartFunc of REAL-NS m,
  REAL-NS n, G be PartFunc of REAL m,REAL n, x be Point of REAL-NS m, y be
Element of REAL m st F=G & x=y & F is_differentiable_in x holds diff(F,x)=diff(
  G,y)
proof
  let m,n be non zero Nat, F be PartFunc of REAL-NS m,REAL-NS n, G
be PartFunc of REAL m,REAL n, x be Point of REAL-NS m, y be Element of REAL m;
  assume that
A1: F=G and
A2: x=y and
A3: F is_differentiable_in x;
A4: the carrier of REAL-NS n = REAL n by REAL_NS1:def 4;
  the carrier of REAL-NS m = REAL m by REAL_NS1:def 4;
  then
A5: diff(F,x) is Function of REAL m,REAL n by A4,LOPBAN_1:def 9;
  G is_differentiable_in y by A1,A2,A3;
  hence thesis by A1,A2,A5,Def8;
end;
