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 Th14:
  f=<>*g & x=y implies (f is_partial_differentiable_in x,i iff g
  is_partial_differentiable_in y,i)
proof
  assume that
A1: f=<>*g and
A2: x=y;
A3: <*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 A3,Th7,Th8;
end;
