reserve T, X, Y for Subset of MC-wff;
reserve p, q, r, s for Element of MC-wff;

theorem Th64:
  ( p 'or' (q 'or' s) ) => ( q 'or' (p 'or' s) ) in IPC-Taut
proof
  set P = p 'or' s, Q = q 'or' s;
A1: P => (q 'or' P) in IPC-Taut by Def14;
  p => P in IPC-Taut by Def14;
  then p => (q 'or' P) in IPC-Taut by A1,Th26;
  then
A2: ((q 'or' P) 'or' p) => ((q 'or' P) 'or' (q 'or' P)) in IPC-Taut by Th63;
  ((q 'or' P) 'or' (q 'or' P)) => (q 'or' P) in IPC-Taut by Th54;
  then
A3: ((q 'or' P) 'or' p) => (q 'or' P) in IPC-Taut by A2,Th26;
  s => P in IPC-Taut by Def14;
  then (q 'or' s) => (q 'or' P) in IPC-Taut by Th63;
  then
A4: (p 'or' Q) => (p 'or' (q 'or' P)) in IPC-Taut by Th63;
  (p 'or' (q 'or' P)) => ((q 'or' P) 'or' p) in IPC-Taut by Th56;
  then (p 'or' Q) => ((q 'or' P) 'or' p) in IPC-Taut by A4,Th26;
  hence thesis by A3,Th26;
end;
