 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 Th123: :: Introduction 'or' to premiss, No.2
  (X \/ {p} |-_IPC r & X \/ {q} |-_IPC r) implies X \/ {p 'or' q} |-_IPC r
proof
  set U = p 'or' q;
  assume A1: (X \/ {p} |-_IPC r & X \/ {q} |-_IPC r); then
A2: X |-_IPC p => r by Th53;
A3: X |-_IPC q => r by A1,Th53;
  X |-_IPC (p => r) => ((q => r) => (U => r)) by Th25; then
  X |-_IPC (q => r) => (U => r) by A2,Th27; then
A6: X |-_IPC U => r by A3,Th27;
A7: {U} |-_IPC U by Th65;
    {U} c= X \/ {U} by XBOOLE_1:7; then
A11: X \/ {U} |-_IPC U by A7,Th66;
   X c= X \/ {U} by XBOOLE_1:7; then
  X \/ {U} |-_IPC U => r by A6,Th66;
  hence thesis by A11,Th27;
end;
