reserve f,g,h for Function,
  A for set;
reserve F for Function,
  B,x,y,y1,y2,z for set;
reserve x,z for object;
reserve X for non empty set,
  Y for set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for non empty set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;

theorem Th38:
  for h being Function of Y,X holds (for z being Element of Y
  holds h.z = F.(f.z,g.z)) implies h = F.:(f,g)
proof
  let h be Function of Y,X;
  assume
A1: for z being Element of Y holds h.z = F.(f.z,g.z);
  now
    let z be Element of Y;
    thus h.z = F.(f.z,g.z) by A1
      .= (F.:(f,g)).z by Th37;
  end;
  hence thesis by FUNCT_2:63;
end;
