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
  p1 is_an_universal_closure_of p implies (X \/ {p} |- q iff X |- p1 => q)
proof
  assume
A1: p1 is_an_universal_closure_of p;
  now
    assume X \/ {p} |- q;
    then
A2: X \/ {p} |- {q} by Th10;
    p |-| p1 by A1,Th38;
    then {p} |-| {p1} by Th29;
    then X \/ {p1} |- {q} by A2,Th26;
    then
A3: X \/ {p1} |- q by Th10;
    p1 is closed by A1;
    hence X |- p1 => q by A3,CQC_THE2:92;
  end;
  hence X \/ {p} |- q implies X |- p1 => q;
  now
    assume X |- p1 => q;
    then X \/ {p1} |- q by Th40;
    then
A4: X \/ {p1} |- {q} by Th10;
    p |-| p1 by A1,Th38;
    then {p} |-| {p1} by Th29;
    then X \/ {p} |- {q} by A4,Th26;
    hence X \/ {p} |- q by Th10;
  end;
  hence thesis;
end;
