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;

theorem Th3:
  X is with_examples implies (X |- Ex(x,p) iff ex y st X |- p.(x,y))
proof
  assume
A1: X is with_examples;
  thus X |- Ex(x,p) implies ex y st X |- p.(x,y)
  proof
    assume X |- Ex(x,p);
    then consider f1 such that
A2: rng f1 c= X and
A3: |- f1^<*Ex(x,p)*> by HENMODEL:def 1;
    consider y such that
A4: X |- 'not' Ex(x,p) 'or' (p.(x,y)) by A1;
    consider f2 such that
A5: rng f2 c= X and
A6: |- f2^<*'not' Ex(x,p) 'or' (p.(x,y))*> by A4,HENMODEL:def 1;
    take y;
A7: |- f1^f2^<*Ex(x,p)*> by A3,HENMODEL:5;
    |- f1^f2^<*'not' Ex(x,p) 'or' (p.(x,y))*> by A6,CALCUL_2:20;
    then
A8: |- f1^f2^<*p.(x,y)*> by A7,Th2;
    rng(f1^f2) = rng f1 \/ rng f2 by FINSEQ_1:31;
    then rng(f1^f2) c= X by A2,A5,XBOOLE_1:8;
    hence thesis by A8,HENMODEL:def 1;
  end;
  thus (ex y st X |- p.(x,y)) implies X |- Ex(x,p)
  proof
    given y such that
A9: X |- p.(x,y);
    consider f1 such that
A10: rng f1 c= X and
A11: |- f1^<*p.(x,y)*> by A9,HENMODEL:def 1;
    |- f1^<*Ex(x,p)*> by A11,CALCUL_1:58;
    hence thesis by A10,HENMODEL:def 1;
  end;
end;
