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;

theorem Th15:
  f = <>*g & x = y & f is_partial_differentiable_in x,i implies
  partdiff(f,x,i).<*1*> = <*partdiff(g,y,i)*>
proof
  assume that
A1: f=<>*g and
A2: x=y and
A3: f is_partial_differentiable_in x,i;
A4: f*reproj(i,x) is_differentiable_in Proj(i,n).x by A3;
A5: <*proj(i,n).y*> = Proj(i,n).x by A2,Def4;
  f*reproj(i,x) =<>*(g*reproj(i,y)) by A1,A2,Th13;
  hence thesis by A4,A5,Th10;
end;
