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;

theorem Th35:
  B is quantifiable & B1 is quantifiable & Sub_All(B,SQ) = Sub_All
  (B1,SQ1) implies B`2 = B1`2 & SQ = SQ1
proof
  assume that
A1: B is quantifiable and
A2: B1 is quantifiable & Sub_All(B,SQ) = Sub_All(B1,SQ1);
  Sub_All(B,SQ) = [All(B`2,(B`1)`1),SQ] by A1,SUBSTUT1:def 24;
  then
A3: [All(B`2,(B`1)`1),SQ] = [All(B1`2,(B1`1)`1),SQ1] by A2,SUBSTUT1:def 24;
  then All(B`2,(B`1)`1) = All(B1`2,(B1`1)`1) by XTUPLE_0:1;
  hence thesis by A3,QC_LANG2:5,XTUPLE_0:1;
end;
