reserve i for Nat, x,y for set;
reserve S for non empty non void ManySortedSign;
reserve X for non-empty ManySortedSet of S;

theorem Th5:
  for f,g being Function st g = f.x holds g.y = f..(x,y)
  proof
    let f,g be Function such that
A1: g = f.x;
A2: f..(x,y) = (uncurry f).(x,y) by FUNCT_6:def 5;
    per cases;
    suppose
      x in dom f & y in dom g; then
A3:   [x,y] in dom uncurry f & f.[x,y]`1 is Function by A1,FUNCT_5:def 2;
      [x,y]`1 = x & [x,y]`2 = y;
      hence g.y = f..(x,y) by A1,A2,A3,FUNCT_5:def 2;
    end;
    suppose
A4:   y nin dom g or x nin dom f; then
A5:   (f.x = 0 or g.y = 0) & dom {} = {}
      by FUNCT_1:def 2;
      now
        assume [x,y] in dom uncurry f; then
        consider a being object, h being Function, b being object such that
A6:     [x,y] = [a,b] & a in dom f & h = f.a & b in dom h by FUNCT_5:def 2;
        a = x & b = y by A6,XTUPLE_0:1;
        hence contradiction by A1,A4,A6;
      end; then
      f..(x,y) = 0 & g.y = 0 by A1,A2,A5,FUNCT_1:def 2;
      hence g.y = f..(x,y);
    end;
  end;
