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:];
reserve a for Element of A;

theorem Th25:
  Suc(f) = All(x,p) & Ant(f) |= Suc(f) implies for y holds Ant(f) |= p.(x,y)
proof
  assume
A1: Suc(f) = All(x,p) & Ant(f) |= Suc(f);
  let y,A,J,v;
  assume J,v |= Ant(f);
  then
A2: J,v |= All(x,p) by A1;
  ex a st v.y = a & J,v.(x|a) |= p
  proof
    take v.y;
    thus thesis by A2,SUBLEMMA:50;
  end;
  hence thesis by Th24;
end;
