reserve A for QC-alphabet;
reserve p, q, r, s, p1, q1 for Element of CQC-WFF(A),
  X, Y, Z, X1, X2 for Subset of CQC-WFF(A),
  h for QC-formula of A,
  x, y for bound_QC-variable of A,
  n for Element of NAT;

theorem Th40:
  X |- p => q implies X \/ {p} |- q
proof
  p in {p} by TARSKI:def 1;
  then p in X \/ {p} by XBOOLE_0:def 3;
  then
A1: X \/ {p} |- p by Th1;
  assume X |- p => q;
  then X \/ {p} |- p => q by Th4,XBOOLE_1:7;
  hence thesis by A1,CQC_THE1:55;
end;
