reserve x,y,z for set;

theorem Th9:
  for S being ManySortedSign, X being ManySortedSet of the carrier
of S for t being non empty Relation, V being ManySortedSubset of X holds V = X
  variables_in t iff for s being set st s in the carrier of S holds V.s = (X.s)
  /\ {a`1 where a is Element of rng t: a`2 = s}
proof
  let S be ManySortedSign, X be ManySortedSet of the carrier of S;
  let t be non empty Relation, V be ManySortedSubset of X;
  hereby
    assume
A1: V = X variables_in t;
    let s be set;
    assume
A2: s in the carrier of S;
    then V.s = (X.s) /\ ((S variables_in t).s) by A1,PBOOLE:def 5;
    hence V.s = (X.s) /\ {a`1 where a is Element of rng t: a`2 = s} by A2,Def2;
  end;
  assume
A3: for s being set st s in the carrier of S holds V.s = (X.s) /\ {a`1
  where a is Element of rng t: a`2 = s};
  now
    let s be object;
    assume
A4: s in the carrier of S;
    hence V.s = (X.s) /\ {a`1 where a is Element of rng t: a`2 = s} by A3
      .= (X.s) /\ ((S variables_in t).s) by A4,Def2;
  end;
  hence thesis by PBOOLE:def 5;
end;
