theorem Th55:
  [:Sgm P " X,Sgm Q " Y:] c= Indices Segm(A,P,Q)
proof
  set SP=Sgm P;
  set SQ=Sgm Q;
  set I=Indices Segm(A,P,Q);
A1: now
    per cases;
    suppose
A2:   card P=0;
      then Seg card P = {};
      then [:Seg card P,Seg card Q:]={} by ZFMISC_1:90;
      hence [:Seg card P,Seg card Q:]=I by A2,MATRIX_0:22;
    end;
    suppose
      card P>0;
      hence [:Seg card P,Seg card Q:]=I by MATRIX_0:23;
    end;
  end;
  dom SQ=Seg card Q by FINSEQ_3:40;
  then
A3: SQ " Y c= Seg card Q by RELAT_1:132;
  dom SP=Seg card P by FINSEQ_3:40;
  then SP " X c= Seg card P by RELAT_1:132;
  hence thesis by A3,A1,ZFMISC_1:96;
end;
