reserve Al for QC-alphabet;
reserve b,c,d for set,
  X,Y for Subset of CQC-WFF(Al),
  i,j,k,m,n for Nat,
  p,p1,q,r,s,s1 for Element of CQC-WFF(Al),
  x,x1,x2,y,y1 for bound_QC-variable of Al,
  A for non empty set,
  J for interpretation of Al, A,
  v for Element of Valuations_in(Al,A),
  f1,f2 for FinSequence of CQC-WFF(Al),
  CX,CY,CZ for Consistent Subset of CQC-WFF(Al),
  JH for Henkin_interpretation of CX,
  a for Element of A,
  t,u for QC-symbol of Al;
reserve L for PATH of q,p,
  F1,F3 for QC-formula of Al,
  a for set;
reserve C,D for Element of [:CQC-WFF(Al),bool bound_QC-variables(Al):];
reserve K,L for Subset of bound_QC-variables(Al);

theorem Th32:
  X |- p & X c= Y implies Y |- p
proof
  assume that
A1: X |- p and
A2: X c= Y;
  consider f being FinSequence of CQC-WFF(Al) such that
A3: rng f c= X and
A4: |- f^<*p*> by A1,HENMODEL:def 1;
  rng f c= Y by A2,A3;
  hence thesis by A4,HENMODEL:def 1;
end;
