theorem Th141:
  variables_in H1 '&' H2 = variables_in H1 \/ variables_in H2
proof
A1: variables_in H1 \/ variables_in H2 = (rng H1 \/ rng H2) \ {0,1,2,3,4} by
XBOOLE_1:42;
A2: rng(H1 '&' H2) = rng (<*3*>^H1) \/ rng H2 by FINSEQ_1:31
    .= rng <*3*> \/ rng H1 \/ rng H2 by FINSEQ_1:31
    .= {3} \/ rng H1 \/ rng H2 by FINSEQ_1:39
    .= {3} \/ (rng H1 \/ rng H2) by XBOOLE_1:4;
  thus variables_in H1 '&' H2 c= variables_in H1 \/ variables_in H2
  proof
    let a be object;
    assume
A3: a in variables_in H1 '&' H2;
    then a <> 3 by Th137;
    then not a in {3} by TARSKI:def 1;
    then
A4: a in rng H1 \/ rng H2 by A2,A3,XBOOLE_0:def 3;
    not a in {0,1,2,3,4} by A3,XBOOLE_0:def 5;
    hence thesis by A1,A4,XBOOLE_0:def 5;
  end;
  thus thesis by A2,A1,XBOOLE_1:7,33;
end;
