 reserve i,n for Nat;
 reserve r for Real;
 reserve ra for Element of F_Real;
 reserve a,b,c for non zero Element of F_Real;
 reserve u,v for Element of TOP-REAL 3;
 reserve p1 for FinSequence of (1-tuples_on REAL);
 reserve pf,uf for FinSequence of F_Real;
 reserve N for Matrix of 3,F_Real;
 reserve K for Field;
 reserve k for Element of K;

theorem Th01:
  1.(F_Real,3) = <* <* 1,0,0 *>,
                    <* 0,1,0 *>,
                    <* 0,0,1 *> *>
  proof
A1: len 1.(F_Real,3) = 3 by MATRIX_0:23;
    1 in Seg 3 & 2 in Seg 3 & 3 in Seg 3 by FINSEQ_1:1;
    then (1.(F_Real,3)).1 = Line(1.(F_Real,3),1) &
         (1.(F_Real,3)).2 = Line(1.(F_Real,3),2) &
         (1.(F_Real,3)).3 = Line(1.(F_Real,3),3) by MATRIX_0:52;
    hence thesis by A1,FINSEQ_1:45,ANPROJ_8:68;
  end;
