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;
