
theorem Th44:
  for u,v,w being Element of TOP-REAL 3 st u in LSeg(v,w) holds
  |[u`1,u`2]| in LSeg ( |[v`1,v`2]| , |[w`1,w`2]| )
  proof
    let u,v,w be Element of TOP-REAL 3;
    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 rv = (1 - r) * v,rw = r * w as Element of TOP-REAL 3;
    rv = |[ (1 - r) * v`1, (1 - r) * v`2, (1 - r) * v`3 ]| &
      rw = |[r * w`1,r * w`2, r * w`3]| by EUCLID_5:7;
    then |[ (1 - r) * v`1 + r * w`1,
      (1 - r) * v`2 + r * w`2,(1 - r) * v`3 + r * w`3 ]| = u by A3,EUCLID_5:6
                                                        .= |[u`1,u`2,u`3]|
                                                          by EUCLID_5:3;
    then
A4: u`1 = (1 - r) * v`1 + r * w`1 &
    u`2 = (1 - r) * v`2 + r  *w`2 by FINSEQ_1:78;
    reconsider u9 = |[u`1,u`2]|, v9 = |[v`1,v`2]|,
               w9 = |[w`1,w`2]| as Element of TOP-REAL 2;
A5: u9`1 = u`1 & u9`2 = u`2 & v9`1 = v`1 & v9`2 = v`2 & w9`1 = w`1 &
    w9`2 = w`2 by EUCLID:52;
    reconsider rv9 = (1 - r) * v9, rw9 = r * w9 as Element of TOP-REAL 2;
    rv9 = |[ (1 - r) * v9`1,(1 - r) * v9`2 ]| &
      rw9 = |[ r * w9`1, r * w9`2 ]| by EUCLID:57;
    then rv9 + rw9 = |[ u`1,u`2 ]| by A4,A5,EUCLID:56;
    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;
