theorem Th119: :: Introduction => to premiss, No.2
  (X |-_IPC p) & (X \/ {r} |-_IPC q) implies X \/ {p => r} |-_IPC q
proof
  assume A1: (X |-_IPC p) & (X \/ {r} |-_IPC q);
   X c= X \/ {p => r} by XBOOLE_1:7; then
A2: X \/ {p => r} |-_IPC p by A1,Th66;
A3: p => r in {p => r} by TARSKI:def 1;
   {p => r} c= X \/ {p => r} by XBOOLE_1:7; then
   p => r in X \/ {p => r} by A3; then
   X \/ {p => r} |-_IPC p => r by Th67; then
A7: X \/ {p => r} |-_IPC r by A2,Th27;
A8: X \/ {r}  c= (X \/ {r}) \/ {p => r} by XBOOLE_1:7;
  (X \/ {r}) \/ {p => r} = (X \/ {p => r}) \/ {r}
      by XBOOLE_1:4; then
  (X \/ {p => r}) \/ {r} |-_IPC q by A1,A8,Th66;
  hence thesis by A7,Th116;
end;
