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 Th24:
  J,v |= p.(x,y) iff ex a st v.y = a & J,v.(x|a) |= p
proof
A1: J,v |= CQC_Sub([p,Sbst(x,y)]) iff J,v.Val_S(v,[p,Sbst(x,y)]) |= [p,Sbst(
  x,y)] by SUBLEMMA:89;
A2: J,v.Val_S(v,[p,Sbst(x,y)]) |= [p,Sbst(x,y)] iff J,v.Val_S(v,[p,Sbst(x,y)
  ]) |= p by Th23;
  Val_S(v,[p,Sbst(x,y)]) = v*(@[p,Sbst(x,y)]`2) by SUBLEMMA:def 1;
  then Val_S(v,[p,Sbst(x,y)]) = v*@Sbst(x,y);
  then
A3: Val_S(v,[p,Sbst(x,y)]) = v*(x .--> y) by SUBSTUT1:def 2;
  y in bound_QC-variables(Al);
  then y in dom v by SUBLEMMA:58;
  then Val_S(v,[p,Sbst(x,y)]) = x .--> v.y by A3,FUNCOP_1:17;
  hence thesis by A1,A2,SUBSTUT2:def 1;
end;
