reserve Al for QC-alphabet;
reserve i,j,k for Nat,
  A,D for non empty set;
reserve f1,f2 for Element of Funcs(Valuations_in(Al,A),BOOLEAN),
  x,x1,y for bound_QC-variable of Al,
  v,v1 for Element of Valuations_in(Al,A);
reserve ll for CQC-variable_list of k,Al;
reserve p,q,s,t for Element of CQC-WFF(Al),
  J for interpretation of Al,A,
  P for QC-pred_symbol of k,Al,
  r for Element of relations_on A;

theorem Th7:
  p = P!ll & r = J.P implies (v*'ll in r iff Valid(p,J).v = TRUE)
proof
  assume that
A1: p = P!ll and
A2: r = J.P;
A3: now
    assume Valid(p,J).v = TRUE;
    then (ll 'in' (J.P)).v <> FALSE by A1,Lm1;
    hence v*'ll in r by A2,Def4;
  end;
  now
    assume v*'ll in r;
    then (ll 'in' (J.P)).v = TRUE by A2,Def4;
    hence Valid(p,J).v = TRUE by A1,Lm1;
  end;
  hence thesis by A3;
end;
