reserve P for Subset of TOP-REAL 2,
  f,f1,f2,g for FinSequence of TOP-REAL 2,
  p,p1,p2,q,q1,q2 for Point of TOP-REAL 2,
  r1,r2,r19,r29 for Real,
  i,j,k,n for Nat;

theorem
  r1<=r2 & r19<=r29 implies [.r1,r2,r19,r29.] = { p: p`1 = r1 & p`2 <=
r29 & p`2 >= r19 or p`1 <= r2 & p`1 >= r1 & p`2 = r29 or p`1 <= r2 & p`1 >= r1
  & p`2 = r19 or p`1 = r2 & p`2 <= r29 & p`2 >= r19}
proof
  assume that
A1: r1<=r2 and
A2: r19<=r29;
  set p1 = |[r1,r19]|, p2 = |[r1,r29]| , p3 = |[r2,r29]|, p4 = |[r2,r19]|;
  set P4 = { p: p`2 = r19& r1 <= p`1 & p`1 <= r2 };
  set P3 = { p: p`1 = r2 & r19<= p`2 & p`2 <= r29};
  set P2 = { p: p`2 = r29& r1 <= p`1 & p`1 <= r2 };
  set P1 = { p: p`1 = r1 & r19<= p`2 & p`2 <= r29};
  set P = { p: p`1 = r1 & p`2 <= r29 & p`2 >= r19 or p`1 <= r2 & p`1 >= r1 & p
`2 = r29 or p`1 <= r2 & p`1 >= r1 & p`2 = r19 or p`1 = r2 & p`2 <= r29 & p`2 >=
  r19};
A3: P = P1 \/ P2 \/ (P3 \/ P4)
  proof
    hereby
      let x be object;
      assume x in P;
      then
      ex p st x = p &( p`1 = r1 & p`2 <= r29 & p`2 >= r19 or p `1 <= r2 &
p`1 >= r1 & p`2 = r29 or p`1 <= r2 & p`1 >= r1 & p`2 = r19 or p`1 = r2 & p`2 <=
      r29 & p`2 >= r19);
      then x in P1 or x in P2 or x in P3 or x in P4;
      then x in P1 \/ P2 or x in P3 \/ P4 by XBOOLE_0:def 3;
      hence x in P1 \/ P2 \/ (P3 \/ P4) by XBOOLE_0:def 3;
    end;
    let x be object;
    assume x in P1 \/ P2 \/ (P3 \/ P4);
    then
A4: x in P1 \/ P2 or x in P3 \/ P4 by XBOOLE_0:def 3;
    per cases by A4,XBOOLE_0:def 3;
    suppose
      x in P1;
      then ex p st x = p & p`1 = r1 & r19<= p`2 & p`2 <= r29;
      hence thesis;
    end;
    suppose
      x in P2;
      then ex p st x = p & p`2 = r29& r1 <= p`1 & p`1 <= r2;
      hence thesis;
    end;
    suppose
      x in P3;
      then ex p st x = p & p`1 = r2 & r19<= p`2 & p`2 <= r29;
      hence thesis;
    end;
    suppose
      x in P4;
      then ex p st x = p & p`2 = r19& r1 <= p`1 & p`1 <= r2;
      hence thesis;
    end;
  end;
  thus [.r1,r2,r19,r29.] = P1 \/ LSeg(p2,p3) \/ (LSeg(p3,p4) \/ LSeg(p4,p1))
  by A2,TOPREAL3:9
    .= P1 \/ P2 \/ (LSeg(p3,p4) \/ LSeg(p4,p1)) by A1,TOPREAL3:10
    .= P1 \/ P2 \/ (P3 \/ LSeg(p4,p1)) by A2,TOPREAL3:9
    .= P by A1,A3,TOPREAL3:10;
end;
