reserve x,y for object, X for set;

theorem Th3:
  for X be set,p,q be ManySortedSet of X st p| (support p) = q| (
  support q) holds p = q
proof
  let X be set,p,q be ManySortedSet of X;
A1: dom (p| (support p)) = dom p /\ (support p) by RELAT_1:61
    .=support p by PRE_POLY:37,XBOOLE_1:28;
A2: dom (q| (support q)) =dom q /\ (support q) by RELAT_1:61
    .=support q by PRE_POLY:37,XBOOLE_1:28;
  assume
A3: p| (support p) = q| (support q);
A4: for x being object st x in X holds p.x=q.x
  proof
    let x being object;
    assume x in X;
    per cases;
    suppose
A5:   x in support p;
      hence p.x=p| (support p).x by FUNCT_1:49
        .=q.x by A3,A1,A2,A5,FUNCT_1:49;
    end;
    suppose
A6:   not x in support p;
      hence p.x = 0 by PRE_POLY:def 7
        .=q.x by A3,A1,A2,A6,PRE_POLY:def 7;
    end;
  end;
A7: dom q = X by PARTFUN1:def 2;
  dom p = X by PARTFUN1:def 2;
  hence thesis by A4,A7,FUNCT_1:2;
end;
