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 Th17:
  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*(1,1) = p1 & M*(1,2) = q1 & M*(1,3) = r1 &
  M*(2,1) = p2 & M*(2,2) = q2 & M*(2,3) = r2 &
  M*(3,1) = p3 & M*(3,2) = q3 & M*(3,3) = r3
  proof
    let p1,p2,p3,q1,q2,q3,r1,r2,r3 be Real;
    assume M = <*<*p1,q1,r1*>,<*p2,q2,r2*>,<*p3,q3,r3*>*>; then
A1: M.1 = <*p1,q1,r1*> & M.2 = <*p2,q2,r2*> &
      M.3 = <*p3,q3,r3*>;
A2: [1,1] in Indices M & [1,2] in Indices M & [1,3] in Indices M &
    [2,1] in Indices M & [2,2] in Indices M &
    [2,3] in Indices M &[3,1] in Indices M & [3,2] in Indices M &
      [3,3] in Indices M by MATRIX_0:24,Th1;
    then ex s be FinSequence of F_Real st s = M.1 & M*(1,1) = s.1
      by MATRIX_0:def 5;
    hence M*(1,1) = p1 by A1;
    ex s be FinSequence of F_Real st s = M.1 & M*(1,2) = s.2
      by A2,MATRIX_0:def 5;
    hence M*(1,2) = q1 by A1;
    ex s be FinSequence of F_Real st s = M.1 & M*(1,3) = s.3
      by A2,MATRIX_0:def 5;
    hence M*(1,3) = r1 by A1;
    ex s be FinSequence of F_Real st s = M.2 & M*(2,1) = s.1
      by A2,MATRIX_0:def 5;
    hence M*(2,1) = p2 by A1;
    ex s be FinSequence of F_Real st s = M.2 & M*(2,2) = s.2
      by A2,MATRIX_0:def 5;
    hence M*(2,2) = q2 by A1;
    ex s be FinSequence of F_Real st s = M.2 & M*(2,3) = s.3
      by A2,MATRIX_0:def 5;
    hence M*(2,3) = r2 by A1;
    ex s be FinSequence of F_Real st s = M.3 & M*(3,1) = s.1
      by A2,MATRIX_0:def 5;
    hence M*(3,1) = p3 by A1;
    ex s be FinSequence of F_Real st s = M.3 & M*(3,2) = s.2
      by A2,MATRIX_0:def 5;
    hence M*(3,2) = q3 by A1;
    ex s be FinSequence of F_Real st s = M.3 & M*(3,3) = s.3
      by A2,MATRIX_0:def 5;
    hence M*(3,3) = r3 by A1;
  end;
