reserve x,y for set;
reserve C,C9,D,D9,E for non empty set;
reserve c for Element of C;
reserve c9 for Element of C9;
reserve d,d1,d2,d3,d4,e for Element of D;
reserve d9 for Element of D9;
reserve i,j for natural Number;
reserve F for Function of [:D,D9:],E;
reserve p,q for FinSequence of D,
  p9,q9 for FinSequence of D9;
reserve f,f9 for Function of C,D,
  h for Function of D,E;
reserve T,T1,T2,T3 for Tuple of i,D;
reserve T9 for Tuple of i, D9;
reserve S for Tuple of j, D;
reserve S9 for Tuple of j, D9;
reserve F,G for BinOp of D;
reserve u for UnOp of D;
reserve H for BinOp of E;

theorem Th39:
  (for d1,d2 holds h.(F.(d1,d2)) = H.(h.d1,h.d2)) implies
    h*(F[:](f,d)) = H[:](h*f,h.d)
proof
  assume
A1: for d1,d2 holds h.(F.(d1,d2)) = H.(h.d1,h.d2);
  reconsider g = C --> d as Function of C,D;
A2: dom h = D & dom(h*f) = C by FUNCT_2:def 1;
  thus h*(F[:](f,d)) = h*(F.:(f,g)) by FUNCT_2:def 1
    .= H.:(h*f,h*g) by A1,Th37
    .= H[:](h*f,h.d) by A2,FUNCOP_1:17;
end;
