theorem Th48:
  i in Seg card P implies Line(Segm(A,P,Seg width A),i) = Line(A, Sgm P.i)
proof
  assume
A1: i in Seg card P;
  set S=Seg width A;
  set sP=Sgm P;
  len Line(A,sP.i)=width A by MATRIX_0:def 7;
  then
A2: dom Line(A,sP.i)=S by FINSEQ_1:def 3;
  Sgm S = idseq width A by FINSEQ_3:48;
  then Line(A,sP.i) * Sgm S = Line(A,sP.i) by A2,RELAT_1:52;
  hence thesis by A1,Th47;
end;
