theorem
  p`1 = q`1 & r in [. proj2.p,proj2.q .] implies |[p`1,r]| in LSeg(p,q)
proof
  assume
A1: p`1 = q`1;
  assume
A2: r in [. proj2.p,proj2.q .];
A3: |[p`1,r]|`2 = r by EUCLID:52;
  proj2.q = q`2 by PSCOMP_1:def 6;
  then
A4: |[p`1,r]|`2 <= q`2 by A2,A3,XXREAL_1:1;
  proj2.p = p`2 by PSCOMP_1:def 6;
  then p`1 = |[p`1,r]|`1 & p`2 <= |[p`1,r]|`2 by A2,A3,EUCLID:52,XXREAL_1:1;
  hence thesis by A1,A4,GOBOARD7:7;
end;
