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
  J,v |= (p '&' q) iff J,v |= p & J,v |= q
proof
A1: now
    assume J,v |= p & J,v |= q;
    then Valid(p,J).v = TRUE & Valid(q,J).v = TRUE;
    then (Valid(p,J).v) '&' (Valid(q,J).v) = TRUE;
    then (Valid(p,J) '&' Valid(q,J)).v = TRUE by MARGREL1:def 20;
    then Valid(p '&' q,J).v = TRUE by Lm1;
    hence J,v |= (p '&' q);
  end;
  now
    assume J,v |= (p '&' q);
    then Valid(p '&' q,J).v = TRUE;
    then (Valid(p,J) '&' Valid(q,J)).v = TRUE by Lm1;
    then (Valid(p,J).v) '&' (Valid(q,J).v) = TRUE by MARGREL1:def 20;
    then Valid(p,J).v = TRUE & Valid(q,J).v = TRUE by MARGREL1:12;
    hence J,v |= p & J,v |= q;
  end;
  hence thesis by A1;
end;
