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;
reserve vS,vS1,vS2 for Val_Sub of A,Al;
reserve B for Element of [:QC-Sub-WFF(Al),bound_QC-variables(Al):],
  SQ for second_Q_comp of B;
reserve B for CQC-WFF-like Element of [:QC-Sub-WFF(Al),
  bound_QC-variables(Al):],
  xSQ for second_Q_comp of [S,x],
  SQ for second_Q_comp of B;
reserve B1 for Element of [:QC-Sub-WFF(Al),bound_QC-variables(Al):];
reserve SQ1 for second_Q_comp of B1;
reserve a for Element of A;

theorem Th79:
  for vS1 being Function holds (for y st y in dom vS1 holds vS1.y
= v.y) & dom vS misses dom vS1 implies for y st y in (dom vS1) \ {x} holds vS1|
  ((dom vS1) \ {x}).y = (v.vS).y
proof
  let vS1 be Function;
  assume that
A1: for y st y in dom vS1 holds vS1.y = v.y and
A2: dom vS misses dom vS1;
  let y such that
A3: y in (dom vS1) \ {x};
  y in dom vS1 /\ ((dom vS1) \ {x}) by A3,XBOOLE_0:def 4;
  then vS1|((dom vS1) \ {x}).y = vS1.y by FUNCT_1:48;
  then
A4: vS1|((dom vS1) \ {x}).y = v.y by A1,A3;
  not y in dom vS by A2,A3,XBOOLE_0:3;
  hence thesis by A4,FUNCT_4:11;
end;
