theorem :: Cut rule
  (X |-_IPC r) & (Y \/ {r} |-_IPC q) implies X \/ Y |-_IPC q
proof
  assume A1: (X |-_IPC r) & (Y \/ {r} |-_IPC q); then
A2: Y |-_IPC r => q by Th53;
A20: X c= X \/ Y & Y c= X \/ Y by XBOOLE_1:7; then
A3: X \/ Y |-_IPC r => q by A2,Th66;
   X \/ Y |-_IPC r by A1,A20,Th66;
   hence thesis by A3,Th27;
end;
