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 Th18:
  M = <*p,q,r*> implies
  M*(1,1) = p`1 & M*(1,2) = p`2 & M*(1,3) = p`3 &
  M*(2,1) = q`1 & M*(2,2) = q`2 & M*(2,3) = q`3 &
  M*(3,1) = r`1 & M*(3,2) = r`2 & M*(3,3) = r`3
  proof
    assume M = <*p,q,r*>;
    then M.1 = p & M.2 = q & M.3 = r; then
A1: M.1 = <*p`1,p`2,p`3*> & M.2 = <*q`1,q`2,q`3*> & M.3 = <*r`1,r`2,r`3*>
      by EUCLID_5:3;
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) = p`1 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) = p`2 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) = p`3 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) = q`1 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) = q`2 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) = q`3 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) = r`1 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) = r`2 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) = r`3 by A1;
  end;
