reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;
reserve a,b,c for Element of F_Real,
          M,N for Matrix of 3,F_Real;
reserve D        for non empty set;
reserve d1,d2,d3 for Element of D;
reserve A        for Matrix of 1,3,D;
reserve B        for Matrix of 3,1,D;

theorem Th30:
  for M being Matrix of 3,F_Real
  for MR being Matrix of 3,REAL
  for v being Element of TOP-REAL 3
  for uf being FinSequence of F_Real
  for ufr being FinSequence of REAL
  for p being FinSequence of (1-tuples_on REAL)
  st p = M * uf & v = M2F p & len uf = 3 & uf = ufr & MR = M
  holds v = MR * ufr
  proof
    let M be Matrix of 3,F_Real;
    let MR be Matrix of 3,REAL;
    let v be Element of TOP-REAL 3;
    let uf be FinSequence of F_Real;
    let ufr be FinSequence of REAL;
    let p be FinSequence of (1-tuples_on REAL);
    assume that
A1: p = M * uf and
A2: v = M2F p and
A3: len uf = 3 and
A4: uf = ufr and
A5: MR = M;
    consider a,b,c,d,e,f,g,h,i be Element of F_Real such that
A6: M = <* <* a,b,c *>,
           <* d,e,f *>,
           <* g,h,i *> *> by PASCAL:3;
    uf is Element of REAL 3 by A3,EUCLID_8:2;
    then uf in REAL 3;
    then uf in TOP-REAL 3 by EUCLID:22;
    then consider t1,t2,t3 be Real such that
A7: uf = <* t1,t2,t3 *> by EUCLID_5:1;
    reconsider x = t1, y = t2, z = t3 as Element of F_Real by XREAL_0:def 1;
    v = <* a * x + b * y + c * z ,
           d * x + e * y + f * z ,
           g * x + h * y + i * z  *> by PASCAL:8,A1,A2,A6,A7;
    hence thesis by A7,A4,A5,A6,PASCAL:9;
  end;
