reserve x,y,z for set;
reserve Q for left-distributive right-distributive complete Lattice-like non
  empty QuantaleStr,
  a, b, c, d for Element of Q;

theorem Th6:
  (a"\/"b) [*] c = (a [*] c) "\/" (b [*] c) & c [*] (a"\/"b) = (c
  [*] a) "\/" (c [*] b)
proof
  set X1 = {d [*] c: d in {a,b}}, X2 = {c [*] d: d in {a,b}};
  now
    let x be object;
    a in {a,b} & b in {a,b} by TARSKI:def 2;
    hence x = a [*] c or x = b [*] c implies x in X1;
    assume x in X1;
    then ex d st x = d [*] c & d in {a,b};
    hence x = a [*] c or x = b [*] c by TARSKI:def 2;
  end;
  then
A1: X1 = {a [*] c, b [*] c} by TARSKI:def 2;
  now
    let x be object;
    a in {a,b} & b in {a,b} by TARSKI:def 2;
    hence x = c [*] a or x = c [*] b implies x in X2;
    assume x in X2;
    then ex d st x = c [*] d & d in {a,b};
    hence x = c [*] a or x = c [*] b by TARSKI:def 2;
  end;
  then
A2: X2 = {c [*] a, c [*] b} by TARSKI:def 2;
A3: a"\/"b = "\/"{a,b} by LATTICE3:43;
  then (a"\/"b) [*] c = "\/"(X1, Q) by Def6;
  hence (a"\/"b) [*] c = (a [*] c) "\/" (b [*] c) by A1,LATTICE3:43;
  c [*] (a"\/"b) = "\/"(X2, Q) by A3,Def5;
  hence thesis by A2,LATTICE3:43;
end;
