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 Th19:
  for p1,p2,p3,q1,q2,q3,r1,r2,r3 being Real holds
  M = <*<*p1,q1,r1*>,<*p2,q2,r2*>,<*p3,q3,r3*>*> implies
  M@ = <*<*p1,p2,p3*>,<*q1,q2,q3*>,<*r1,r2,r3*>*>
  proof
    let p1,p2,p3,q1,q2,q3,r1,r2,r3 be Real;
    assume
A1: M = <*<*p1,q1,r1*>,<*p2,q2,r2*>,<*p3,q3,r3*>*>;
    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) = p1 & M*(2,1) = p2 & M*(3,1) = p3 &
    M*(1,2) = q1 & M*(2,2) = q2 & M*(3,2) = q3 &
    M*(1,3) = r1 & M*(2,3) = r2 & M*(3,3) = r3 by A1,Th17;
    hence thesis by A2,MATRIXR2:37;
  end;
