theorem Th118:
  P c= Seg m implies lines Segm(M,P,Seg n) c= lines M
proof
  set S=Segm(M,P,Seg n);
  assume
A1: P c= Seg m;
A2: rng Sgm P = P by FINSEQ_1:def 14;
  let x be object;
  assume x in lines S;
  then consider i such that
A3: i in Seg card P and
A4: x = Line(S,i) by Th103;
  Seg m <> {} by A1,A3;
  then m <> 0;
  then width M=n by Th1;
  then
A5: Line(S,i)=Line(M,Sgm P.i) by A3,Th48;
  dom Sgm P=Seg card P by FINSEQ_3:40;
  then Sgm P.i in rng Sgm P by A3,FUNCT_1:def 3;
  hence thesis by A1,A4,A2,A5,Th103;
end;
