 reserve i,j,n,k,l for Nat;
 reserve T,S,X,Y,Z for Subset of MC-wff;
 reserve p,q,r,t,F,H,G for Element of MC-wff;
 reserve s,U,V for MC-formula;
reserve f,g for FinSequence of [:MC-wff,Proof_Step_Kinds_IPC:];
 reserve X,T for Subset of MC-wff;
 reserve F,G,H,p,q,r,t for Element of MC-wff;
 reserve s,h for MC-formula;
 reserve f for FinSequence of [:MC-wff,Proof_Step_Kinds_IPC:];
 reserve i,j for Element of NAT;
 reserve F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,G for MC-formula;
 reserve x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x for Element of MC-wff;
reserve x1,x2,x3,x4,x5,x6,x7,x8,x9,x10 for object;

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;
