reserve Al for QC-alphabet;
reserve a,b,c,d for object,
  i,j,k,m,n for Nat,
  p,q,r for Element of CQC-WFF(Al),
  x,y,y0 for bound_QC-variable of Al,
  X for Subset of CQC-WFF(Al),
  A for non empty set,
  J for interpretation of Al,A,
  v,w for Element of Valuations_in(Al,A),
  Sub for CQC_Substitution of Al,
  f,f1,g,h,h1 for FinSequence of CQC-WFF(Al);
reserve fin,fin1 for FinSequence;
reserve PR,PR1 for FinSequence of [:set_of_CQC-WFF-seq(Al),Proof_Step_Kinds:];

theorem Th20:
  Ant(f) = Ant(g) & Ant(f) |= Suc(f) & Ant(g) |= Suc(g) implies
  Ant(f) |= (Suc(f)) '&' (Suc(g))
proof
  assume
A1: Ant(f) = Ant(g) & Ant(f) |= Suc(f) & Ant(g) |= Suc(g);
  let A,J,v;
  assume J,v |= Ant(f);
  then J,v |= Suc(f) & J,v |= Suc(g) by A1;
  hence thesis by VALUAT_1:18;
end;
