theorem Th103:
  x in lines M iff ex i st i in Seg m & x = Line(M,i)
proof
  thus x in lines M implies ex i st i in Seg m & x = Line(M,i)
  proof
    assume x in lines M;
    then consider i be object such that
A1: i in dom M and
A2: M.i=x by FUNCT_1:def 3;
A3: len M=m by MATRIX_0:def 2;
    reconsider i as Element of NAT by A1;
A4: dom M=Seg len M by FINSEQ_1:def 3;
    then M.i=Line(M,i) by A1,A3,MATRIX_0:52;
    hence thesis by A1,A2,A4,A3;
  end;
  given i such that
A5: i in Seg m and
A6: x = Line(M,i);
A7: len M=m by MATRIX_0:def 2;
  dom M=Seg len M by FINSEQ_1:def 3;
  then M.i in rng M by A5,A7,FUNCT_1:def 3;
  hence thesis by A5,A6,MATRIX_0:52;
end;
