reserve Al for QC-alphabet;
reserve a,b,c,d for object,
  i,k,n for Nat,
  p,q for Element of CQC-WFF(Al),
  x,y,y1 for bound_QC-variable of Al,
  A for non empty set,
  J for interpretation of Al,A,
  v,w for Element of Valuations_in(Al,A),
  f,g for Function,
  P,P9 for QC-pred_symbol of k,Al,
  ll,ll9 for CQC-variable_list of k,Al,
  l1 for FinSequence of QC-variables(Al),
  Sub,Sub9,Sub1 for CQC_Substitution of Al,
  S,S9,S1,S2 for Element of CQC-Sub-WFF(Al),
  s for QC-symbol of Al;

theorem Th2:
  for vS1 being Function st x in dom vS1 holds (vS1|((dom vS1) \ {x
  })) +* (x .--> vS1.x) = vS1
proof
  let vS1 be Function;
  assume x in dom vS1;
  then (dom vS1 \ {x}) \/ {x} = dom vS1 \/ {x} & {x} c= dom vS1 by XBOOLE_1:39
,ZFMISC_1:31;
  then (dom vS1 \ {x}) \/ {x} = dom vS1 by XBOOLE_1:12;
  hence thesis by FUNCT_7:14;
end;
