reserve A for QC-alphabet;
reserve sq for FinSequence,
  x,y,z for bound_QC-variable of A,
  p,q,p1,p2,q1 for Element of QC-WFF(A);

theorem Th2:
  p '&' q = p1 '&' q1 implies p = p1 & q = q1
proof
  assume
A1: p '&' q = p1 '&' q1;
  <*[2,0]*>^@p^@q = <*[2,0]*>^(@p^@q) & <*[2,0]*>^@p1^@q1 = <*[2,0]*>^(@p1
  ^@q1 ) by FINSEQ_1:32;
  then
A2: @p^@q = @p1^@q1 by A1,FINSEQ_1:33;
  then
A3: len @p1 <= len @p implies ex sq st @p = @p1^sq by FINSEQ_1:47;
A4: len @p <= len @p1 implies ex sq st @p1 = @p^sq by A2,FINSEQ_1:47;
  hence p = p1 by A3,QC_LANG1:13;
  (ex sq st @p1 = @p^sq) implies p1 = p by QC_LANG1:13;
  hence thesis by A1,A3,A4,FINSEQ_1:33,QC_LANG1:13;
end;
