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 Th56:
  Line(1.(F_Real,3),1) = <* 1,0,0 *> &
  Line(1.(F_Real,3),2) = <* 0,1,0 *> &
  Line(1.(F_Real,3),3) = <* 0,0,1 *>
  proof
    now
      thus len Line(1.(F_Real,3),1) = width 1.(F_Real,3) by MATRIX_0:def 7
                                   .= 3 by MATRIX_0:23;
      [1,1] in Indices(1.(F_Real,3)) by MATRIX_0:23,Th1;
      then Line(1.(F_Real,3),1).1 = 1.F_Real by MATRIX_3:7
                                 .= 1 by STRUCT_0:def 7;
      hence Line(1.(F_Real,3),1).1 = 1;
      [1,2] in Indices(1.(F_Real,3)) by MATRIX_0:23,Th1;
      then Line(1.(F_Real,3),1).2 = 0.F_Real by MATRIX_3:8
                                 .= 0 by STRUCT_0:def 6;
      hence Line(1.(F_Real,3),1).2 = 0;
      [1,3] in Indices(1.(F_Real,3)) by MATRIX_0:23,Th1;
      then Line(1.(F_Real,3),1).3 = 0.F_Real by MATRIX_3:8
                                 .= 0 by STRUCT_0:def 6;
      hence Line(1.(F_Real,3),1).3 = 0;
    end;
    hence Line(1.(F_Real,3),1) = <* 1,0,0 *> by FINSEQ_1:45;
    now
      thus len Line(1.(F_Real,3),2) = width 1.(F_Real,3) by MATRIX_0:def 7
                                   .= 3 by MATRIX_0:23;
      [2,1] in Indices(1.(F_Real,3)) by MATRIX_0:23,Th1;
      then Line(1.(F_Real,3),2).1 = 0.F_Real by MATRIX_3:8
                                 .= 0 by STRUCT_0:def 6;
      hence Line(1.(F_Real,3),2).1 = 0;
      [2,2] in Indices(1.(F_Real,3)) by MATRIX_0:23,Th1;
      then Line(1.(F_Real,3),2).2 = 1.F_Real by MATRIX_3:7
                                 .= 1 by STRUCT_0:def 7;
      hence Line(1.(F_Real,3),2).2 = 1;
      [2,3] in Indices(1.(F_Real,3)) by MATRIX_0:23,Th1;
      then Line(1.(F_Real,3),2).3 = 0.F_Real by MATRIX_3:8
                                 .= 0 by STRUCT_0:def 6;
      hence Line(1.(F_Real,3),2).3 = 0;
    end;
    hence Line(1.(F_Real,3),2) = <* 0,1,0 *> by FINSEQ_1:45;
    now
      thus len Line(1.(F_Real,3),3) = width 1.(F_Real,3) by MATRIX_0:def 7
                                   .= 3 by MATRIX_0:23;
      [3,1] in Indices(1.(F_Real,3)) by MATRIX_0:23,Th1;
      then Line(1.(F_Real,3),3).1 = 0.F_Real by MATRIX_3:8
                                 .=  0 by STRUCT_0:def 6;
      hence Line(1.(F_Real,3),3).1 = 0;
      [3,2] in Indices(1.(F_Real,3)) by MATRIX_0:23,Th1;
      then Line(1.(F_Real,3),3).2 = 0.F_Real by MATRIX_3:8
                                 .= 0 by STRUCT_0:def 6;
      hence Line(1.(F_Real,3),3).2 = 0;
      [3,3] in Indices(1.(F_Real,3)) by MATRIX_0:23,Th1;
      then Line(1.(F_Real,3),3).3 = 1.F_Real by MATRIX_3:7
                                 .= 1 by STRUCT_0:def 7;
      hence Line(1.(F_Real,3),3).3 = 1;
    end;
    hence Line(1.(F_Real,3),3) = <* 0,0,1 *> by FINSEQ_1:45;
  end;
