theorem
  p`2 = q`2 & r in [. proj1.p,proj1.q .] implies |[r,p`2]| in LSeg(p,q)
proof
  assume
A1: p`2 = q`2;
  assume
A2: r in [. proj1.p,proj1.q .];
A3: |[r,p`2]|`1 = r by EUCLID:52;
  proj1.q = q`1 by PSCOMP_1:def 5;
  then
A4: |[r,p`2]|`1 <= q`1 by A2,A3,XXREAL_1:1;
  proj1.p = p`1 by PSCOMP_1:def 5;
  then p`2 = |[r,p`2]|`2 & p`1 <= |[r,p`2]|`1 by A2,A3,EUCLID:52,XXREAL_1:1;
  hence thesis by A1,A4,GOBOARD7:8;
end;
