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;
reserve pf for FinSequence of D;
reserve PQR for Matrix of 3,F_Real;
reserve R for Ring;

theorem Th74:
  for N being Matrix of 3,R
  for p being FinSequence of R st len p = 3 holds
  N * (<*p*>@) is (3,1)-size
  proof
    let N be Matrix of 3,R;
    let p be FinSequence of R;
    assume
A1: len p = 3;
    then
A2: width <*p*> = 3 by MATRIX_0:23; then
A3: width N = width (<*p*>) by MATRIX_0:24
           .= len (<*p*>@) by MATRIX_0:def 6;
    now
      len (N * (<*p*>@)) = len N by A3,MATRIX_3:def 4;
      hence
A4:   len (N * <*p*>@) = 3 by MATRIX_0:24;
      thus for pf be FinSequence of R st pf in rng (N * <*p*>@) holds
       len pf = 1
      proof
        let pf be FinSequence of R;
        assume
A5:     pf in rng (N * <*p*>@);
A6:     len <*p*> = 1 by MATRIX_0:23;
A7:     width <*p*> = 3 by A1,MATRIX_0:23;
A8:     width N = width (<*p*>) by A2,MATRIX_0:24
               .= len (<*p*>@) by MATRIX_0:def 6;
A9:     width (<*p*>@) = len (<*p*>@@) by MATRIX_0:def 6
                      .= 1 by A6,A7,MATRIX_0:57;
        consider s be FinSequence such that
A10:    s in rng(N * (<*p*>@)) and
A11:    len s = width(N * (<*p*>@)) by A4,MATRIX_0:def 3;
        consider n0 be Nat such that
A12:    for x be object st x in rng (N * (<*p*>@)) ex s be FinSequence st
          s = x & len s = n0 by MATRIX_0:def 1;
A13:    ex s0 be FinSequence st s0 = pf & len s0 = n0 by A12,A5;
        ex s1 be FinSequence st s1 = s & len s1 = n0 by A10,A12;
        hence thesis by A9,A8,MATRIX_3:def 4,A11,A13;
    end;
  end;
  hence thesis by MATRIX_0:def 2;
end;
