
theorem Th65:
  for S being LATTICE st for X being Subset of S st ex_sup_of X,S
  for x being Element of S holds x"/\"("\/"(X,S)) = "\/"({x"/\" y where y is
  Element of S: y in X},S) holds S is distributive
proof
  let S be LATTICE such that
A1: for X being Subset of S st ex_sup_of X,S for x being Element of S
  holds x"/\"("\/"(X,S)) = "\/"({x"/\"y where y is Element of S: y in X},S);
  let x,y,z be Element of S;
  set Y = {x"/\"s where s is Element of S: s in {y,z}};
A2: ex_sup_of {y,z},S by YELLOW_0:20;
  now
    let t be object;
    hereby
      assume t in Y;
      then ex s being Element of S st t = x"/\"s & s in {y,z};
      hence t = x"/\"y or t = x"/\"z by TARSKI:def 2;
    end;
    assume
A3: t = x"/\"y or t = x"/\"z;
    per cases by A3;
    suppose
A4:   t = x"/\"y;
      y in {y,z} by TARSKI:def 2;
      hence t in Y by A4;
    end;
    suppose
A5:   t = x"/\"z;
      z in {y,z} by TARSKI:def 2;
      hence t in Y by A5;
    end;
  end;
  then
A6: Y = {x"/\"y,x"/\"z} by TARSKI:def 2;
  thus x "/\" (y "\/" z) = x "/\" (sup {y,z}) by YELLOW_0:41
    .= "\/"({x"/\"y,x"/\"z},S) by A1,A6,A2
    .= (x "/\" y) "\/" (x "/\" z) by YELLOW_0:41;
end;
