
theorem Th45:
  for u,v,w being Element of TOP-REAL 2 st u in LSeg(v,w) holds
  |[u`1,u`2,1]| in LSeg(|[v`1,v`2,1]|,|[w`1,w`2,1]|)
  proof
    let u,v,w be Element of TOP-REAL 2;
    assume u in LSeg(v,w);
    then consider r be Real such that
A1: 0 <= r and
A2: r <= 1 and
A3: u = (1 - r) * v + r * w by RLTOPSP1:76;
    reconsider u9 = |[u`1,u`2,1]|,
    v9 = |[v`1,v`2,1]|,
    w9 = |[w`1,w`2,1]| as Element of TOP-REAL 3;
    reconsider rv = (1 - r) * v,
    rw = r * w as Element of REAL 2 by EUCLID:22;
A4: rv.1 = (1 - r) * v.1 & rv.2 = (1 - r) * v.2 &
      rw.1 = r * w.1 & rw.2 = r * w.2 by RVSUM_1:44;
A5: u`2 = (1 - r) * v.2 + r * w.2 by A4,A3,RVSUM_1:11;
    reconsider rv9 = (1 - r) * v9,rw9 = r * w9 as Element of TOP-REAL 3;
    u9 = (1 - r) * v9 + r * w9
    proof
      u9 = |[(1 - r) * v.1 + r * w.1,
             (1 - r) * v.2 + r * w.2, (1 - r) * 1 + r * 1]|
             by A5,A3,A4,RVSUM_1:11
        .= |[(1 - r) * v.1,(1 - r) * v.2,(1 - r) * 1]|
             + |[r * w.1, r * w.2, r * 1]| by EUCLID_5:6
        .= (1 - r) * |[v.1,v.2,1]|
             + |[r * w.1, r * w.2, r * 1]| by EUCLID_5:8
        .= (1 - r) * |[v`1,v`2,1]|
             + r * |[w`1,w`2,1]| by EUCLID_5:8;
      hence thesis;
    end;
    then u9 in {(1 - r) * v9 + r * w9 where r is Real: 0 <= r & r <= 1}
      by A1,A2;
    hence thesis by RLTOPSP1:def 2;
  end;
