theorem Th50:
  j in Seg card Q implies Col(Segm(A,Seg len A,Q),j) = Col(A,Sgm Q .j)
proof
  assume
A1: j in Seg card Q;
  set S=Seg len A;
  set sQ=Sgm Q;
  len Col(A,sQ.j)=len A by MATRIX_0:def 8;
  then
A2: dom Col(A,sQ.j)=S by FINSEQ_1:def 3;
  Sgm S = idseq len A by FINSEQ_3:48;
  then Col(A,sQ.j) * Sgm S = Col(A,sQ.j) by A2,RELAT_1:52;
  hence thesis by A1,Th49;
end;
