reserve Al for QC-alphabet;
reserve i,j,k for Nat,
  A,D for non empty set;
reserve f1,f2 for Element of Funcs(Valuations_in(Al,A),BOOLEAN),
  x,x1,y for bound_QC-variable of Al,
  v,v1 for Element of Valuations_in(Al,A);
reserve ll for CQC-variable_list of k,Al;
reserve p,q,s,t for Element of CQC-WFF(Al),
  J for interpretation of Al,A,
  P for QC-pred_symbol of k,Al,
  r for Element of relations_on A;
reserve u,w,z for Element of BOOLEAN;
reserve w,v2 for Element of Valuations_in(Al,A),
  z for bound_QC-variable of Al;
reserve u,w for Element of Valuations_in(Al,A);
reserve s9 for QC-formula of Al;

theorem Th36:
  J,v |= (p => q) => ('not'(q '&' t) => 'not'(p '&' t))
proof
  p => q = 'not'(p '&' 'not' q) & 'not'(q '&' t) => 'not'(p '&' t) = 'not'
  ( 'not'(q '&' t) '&' 'not' 'not' (p '&' t)) by QC_LANG2:def 2;
  then
A1: Valid((p => q) => ('not'(q '&' t) => 'not'(p '&' t)), J).v = Valid('not'
('not'(p '&' 'not' q) '&' 'not'('not'('not'(q '&' t) '&' 'not' 'not'(p '&' t)))
  ), J).v by QC_LANG2:def 2
    .= 'not'(Valid('not'(p '&' 'not' q) '&' 'not'('not'('not'(q '&' t) '&'
  'not' 'not'(p '&' t))), J).v) by Th10
    .= 'not'((Valid('not'(p '&' 'not' q), J).v) '&' (Valid('not'('not'('not'
  (q '&' t) '&' 'not' 'not' (p '&' t))), J).v)) by Th12;
A2: Valid('not'(p '&' 'not' q), J).v = 'not'(Valid(p '&' 'not' q, J).v) by Th10
    .= 'not'((Valid(p, J).v) '&' (Valid('not' q, J).v)) by Th12
    .= 'not'((Valid(p, J).v) '&'('not'(Valid(q, J).v))) by Th10;
  Valid('not'('not'('not'(q '&' t) '&' 'not' 'not'(p '&' t))), J).v =
  'not'(Valid('not'('not'(q '&' t) '&' 'not' 'not' (p '&' t)), J).v) by Th10
    .= 'not' 'not'(Valid('not'(q '&' t) '&' 'not' 'not' (p '&' t), J).v) by
Th10
    .= (Valid('not'(q '&' t),J).v) '&' (Valid('not' 'not' (p '&' t), J).v)
  by Th12
    .= 'not'(Valid(q '&' t,J).v) '&' (Valid('not' 'not' (p '&' t), J).v) by
Th10
    .= 'not'(Valid(q '&' t,J).v) '&'('not'(Valid('not' (p '&' t), J).v)) by
Th10
    .= 'not'(Valid(q '&' t,J).v) '&'('not' 'not' (Valid(p '&' t, J).v)) by Th10
    .= 'not'((Valid(q,J).v) '&' (Valid(t,J).v)) '&' (Valid(p '&' t, J).v) by
Th12
    .= 'not'((Valid(q,J).v) '&' (Valid(t,J).v)) '&' ((Valid(p,J).v) '&' (
  Valid(t,J).v)) by Th12;
  hence
  Valid((p => q) => ('not'(q '&' t) => 'not'(p '&' t)), J).v = TRUE by A1,A2
,Lm2;
end;
