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;
reserve pf for FinSequence of D;
reserve PQR for Matrix of 3,F_Real;

theorem Th73:
  for M being Matrix of 3,1,D holds Line(M,1) = <* M*(1,1) *> &
  Line(M,2) = <* M*(2,1) *> & Line(M,3) = <* M*(3,1) *>
  proof
    let M be Matrix of 3,1,D;
A1: len M = 3 & width M = 1 & Indices M = [:Seg 3,Seg 1:] by MATRIX_0:23;
    now
A2:   len Line(M,1) = width M by MATRIX_0:def 7
                   .= 1 by MATRIX_0:23;
      dom <* M*(1,1) *> = Seg 1 by FINSEQ_1:def 8;
      hence dom Line(M,1) = dom <* M*(1,1) *> by A2,FINSEQ_1:def 3;
      thus for x be object st x in dom Line(M,1) holds
        (Line(M,1)).x = ( <* M*(1,1) *>).x
      proof
        let x be object;
        assume x in dom Line(M,1);
        then x in {1} by A2,FINSEQ_1:def 3,FINSEQ_1:2; then
A3:     x = 1 by TARSKI:def 1;
        (Line(M,1)).1 = M*(1,1) by A1,FINSEQ_1:1,MATRIX_0:def 7
                     .= ( <* M*(1,1) *>).1;
        hence thesis by A3;
      end;
    end;
    hence Line(M,1) = <* M*(1,1) *> by FUNCT_1:def 11;
    now
A4:   len Line(M,2) = width M by MATRIX_0:def 7
                   .= 1 by MATRIX_0:23;
      dom <* M*(2,1) *> = Seg 1 by FINSEQ_1:def 8;
      hence dom Line(M,2) = dom <* M*(2,1) *> by A4,FINSEQ_1:def 3;
      thus for x be object st x in dom Line(M,2) holds
        (Line(M,2)).x = ( <* M*(2,1) *>).x
      proof
        let x be object;
        assume x in dom Line(M,2);
        then x in {1} by A4,FINSEQ_1:def 3,FINSEQ_1:2; then
A5:     x = 1 by TARSKI:def 1;
        (Line(M,2)).1 = M*(2,1) by A1,FINSEQ_1:1,MATRIX_0:def 7
                     .= ( <* M*(2,1) *>).1;
        hence thesis by A5;
      end;
    end;
    hence Line(M,2) = <* M*(2,1) *> by FUNCT_1:def 11;
    now
A6:   len Line(M,3) = width M by MATRIX_0:def 7
                   .= 1 by MATRIX_0:23;
      dom <* M*(3,1) *> = Seg 1 by FINSEQ_1:def 8;
      hence dom Line(M,3) = dom <* M*(3,1) *> by A6,FINSEQ_1:def 3;
      thus for x be object st x in dom Line(M,3) holds
        (Line(M,3)).x = ( <* M*(3,1) *>).x
      proof
        let x be object;
        assume x in dom Line(M,3);
        then x in {1} by A6,FINSEQ_1:def 3,FINSEQ_1:2; then
A7:     x = 1 by TARSKI:def 1;
        (Line(M,3)).1 = M*(3,1) by A1,FINSEQ_1:1,MATRIX_0:def 7
                     .= ( <* M*(3,1) *>).1;
        hence thesis by A7;
      end;
    end;
    hence Line(M,3) = <* M*(3,1) *> by FUNCT_1:def 11;
  end;
