reserve a,b,c,d,e,f for Real,
        k,m for Nat,
        D for non empty set,
        V for non trivial RealLinearSpace,
        u,v,w for Element of V,
        p,q,r for Element of ProjectiveSpace(V);
reserve o,p,q,r,s,t for Point of TOP-REAL 3,
        M for Matrix of 3,F_Real;

theorem Th21:
  lines M = {Line(M,1),Line(M,2),Line(M,3)}
  proof
A1: lines M c= {Line(M,1),Line(M,2),Line(M,3)}
    proof
      let x be object;
      assume x in lines M;
      then consider i be Nat such that
A2:   i in Seg 3 and
A3:   x = Line(M,i) by MATRIX13:103;
      i = 1 or i = 2 or i = 3 by A2,FINSEQ_3:1,ENUMSET1:def 1;
      hence thesis by A3,ENUMSET1:def 1;
    end;
    {Line(M,1),Line(M,2),Line(M,3)} c= lines M
    proof
      let x be object;
      assume x in {Line(M,1),Line(M,2),Line(M,3)}; then
A4:   x = Line(M,1) or x = Line(M,2) or x = Line(M,3) by ENUMSET1:def 1;
      1 in Seg 3 & 2 in Seg 3 & 3 in Seg 3 by FINSEQ_1:1;
      hence thesis by A4,MATRIX13:103;
    end;
    hence thesis by A1;
  end;
