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 Th10:
  JH,valH(Al) |= Ex(x,p) iff ex y st JH,valH(Al) |= p.(x,y)
proof
  thus JH,valH(Al) |= Ex(x,p) implies ex y st JH,valH(Al) |= p.(x,y)
  proof
    assume JH,valH(Al) |= Ex(x,p);
    then consider x1 being Element of HCar(Al) such that
A1: JH,(valH(Al)).(x|x1) |= p by Th9;
A2: HCar(Al) = bound_QC-variables(Al) by HENMODEL:def 4;
    valH(Al) = id bound_QC-variables(Al) by HENMODEL:def 6;
    then rng valH(Al) = bound_QC-variables(Al);
    then consider b being object such that
A3: b in dom valH(Al) and
A4: (valH(Al)).b = x1 by A2,FUNCT_1:def 3;
    reconsider y = b as bound_QC-variable of Al by A3;
    take y;
    thus thesis by A1,A4,CALCUL_1:24;
  end;
  thus (ex y st JH,valH(Al) |= p.(x,y)) implies JH,valH(Al) |= Ex(x,p)
  proof
    given y such that
A5: JH,valH(Al) |= p.(x,y);
    ex x1 being Element of HCar(Al) st ( (valH(Al)).y = x1)&( JH,(valH(Al)).(x|
    x1) |= p) by A5,CALCUL_1:24;
    hence thesis by Th9;
  end;
end;
