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 Th20:
  M = <* p,q,r *> implies
  M@ = <*<*p`1,q`1,r`1*>,<*p`2,q`2,r`2*>,<*p`3,q`3,r`3*>*>
  proof
    assume
A1: M = <* p,q,r *>;
    Indices M = [:Seg 3,Seg 3:] by MATRIX_0:24; then
A2: M@*(1,1) = M*(1,1) & M@*(1,2) = M*(2,1) & M@*(1,3) = M*(3,1) &
      M@*(2,1) = M*(1,2) & M@*(2,2) = M*(2,2) & M@*(2,3) = M*(3,2) &
      M@*(3,1) = M*(1,3) & M@*(3,2) = M*(2,3) & M@*(3,3) = M*(3,3)
      by MATRIX_0:def 6,Th1;
    M*(1,1) = p`1 & M*(2,1) = q`1 & M*(3,1) = r`1 &
    M*(1,2) = p`2 & M*(2,2) = q`2 & M*(3,2) = r`2 &
    M*(1,3) = p`3 & M*(2,3) = q`3 & M*(3,3) = r`3 by A1,Th18;
    hence thesis by A2,MATRIXR2:37;
  end;
