reserve A for QC-alphabet;
reserve X,T for Subset of CQC-WFF(A);
reserve F,G,H,p,q,r,t for Element of CQC-WFF(A);
reserve s,h for QC-formula of A;
reserve x,y for bound_QC-variable of A;
reserve f for FinSequence of [:CQC-WFF(A),Proof_Step_Kinds:];
reserve i,j for Element of NAT;

theorem Th88:
  Cn({p} \/ {q}) = Cn({p '&' q})
proof
  for t holds t in Cn({p} \/ {q}) iff for T st T is being_a_theory & {p
  '&' q} c= T holds t in T
  proof
    let t;
    thus t in Cn({p} \/ {q}) implies for T st T is being_a_theory & {p '&' q}
    c= T holds t in T
    proof
      assume
A1:   t in Cn({p} \/ {q});
      let T;
      assume that
A2:   T is being_a_theory and
A3:   {p '&' q} c= T;
A4:   p '&' q in T & TAUT(A) c= T by A2,A3,CQC_THE1:38,ZFMISC_1:31;
      p '&' q => q in TAUT(A) by PROCAL_1:21;
      then q in T by A2,A4;
      then
A5:   {q} c= T by ZFMISC_1:31;
      p '&' q => p in TAUT(A) by PROCAL_1:19;
      then p in T by A2,A4;
      then {p} c= T by ZFMISC_1:31;
      then {p} \/ {q} c= T by A5,XBOOLE_1:8;
      hence thesis by A1,A2,CQC_THE1:def 2;
    end;
    thus (for T st T is being_a_theory & {p '&' q} c= T holds t in T) implies
    t in Cn({p} \/ {q})
    proof
      assume
A6:   for T st T is being_a_theory & {p '&' q} c= T holds t in T;
      for T st T is being_a_theory & {p} \/ {q} c= T holds t in T
      proof
        let T;
        assume that
A7:     T is being_a_theory and
A8:     {p} \/ {q} c= T;
        {p} c= {p} \/ {q} by XBOOLE_1:7;
        then {p} c= T by A8,XBOOLE_1:1;
        then
A9:     p in T by ZFMISC_1:31;
        {q} c= {p} \/ {q} by XBOOLE_1:7;
        then {q} c= T by A8,XBOOLE_1:1;
        then
A10:    q in T by ZFMISC_1:31;
        p => (q => p '&' q) in TAUT(A) & TAUT(A) c= T
          by A7,CQC_THE1:38,PROCAL_1:28;
        then (q => p '&' q) in T by A7,A9;
        then p '&' q in T by A7,A10;
        then {p '&' q} c= T by ZFMISC_1:31;
        hence thesis by A6,A7;
      end;
      hence thesis by CQC_THE1:def 2;
    end;
  end;
  hence thesis by CQC_THE1:def 2;
end;
