reserve S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem Th8:
  for X,Y be RealNormSpace,
        Z be non empty set,
        f be PartFunc of [:X,Y:],Z,
        I be Function of [:Y,X:],[:X,Y:]
   st ( for y be Point of Y, x be Point of X
        holds I.(y,x) = [x,y] )
  holds
    dom(f*I) = I"(dom f)
  & for x be Point of X, y be Point of Y
    holds (f*I).(y,x) = f.(x,y)
  proof
    let X,Y be RealNormSpace,
          Z be non empty set,
          f be PartFunc of [:X,Y:],Z,
          I be Function of [:Y,X:],[:X,Y:];
    assume
    A1: for x be Point of Y,y be Point of X
        holds I.(x,y) = [y,x];

    for w be object holds
    w in dom(f*I) iff w in I"(dom f)
    proof
      let w be object;
      w in dom(f*I) iff w in dom I & I.w in dom f by FUNCT_1:11;
      hence w in dom (f*I) iff w in I"(dom f) by FUNCT_1:def 7;
    end;
    hence dom(f*I) = I"(dom f) by TARSKI:2;

    let x be Point of X, y be Point of Y;
    [y,x] in the carrier of [:Y,X:]; then
    [y,x] in dom I by FUNCT_2:def 1;
    hence (f*I).(y,x) = f.(I.(y,x)) by FUNCT_1:13
                    .= f.(x,y) by A1;
  end;
