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
  for p being FinSequence of 1-tuples_on REAL st len p = 3 holds
  ColVec2Mx M2F p = p
  proof
    let p be FinSequence of 1-tuples_on REAL;
    assume
A1: len p = 3;
    then
A2: M2F p = <* (p.1).1,(p.2).1,(p.3).1 *> by ANPROJ_8:def 2; then
A3: len (M2F p) = 3 by FINSEQ_1:45;
A4: len (M2F p) > 0 by A2,FINSEQ_1:45; then
A5: len(ColVec2Mx M2F p) = len (M2F p) by MATRIXR1:def 9
                        .= 3 by A2,FINSEQ_1:45;
    width (ColVec2Mx M2F p) = 1 & ColVec2Mx M2F p is Matrix of REAL
      by A4,MATRIXR1:def 9; then
A6: ColVec2Mx M2F p is Matrix of 3,1,F_Real by A5,MATRIX_0:20;
    set A = ColVec2Mx M2F p;
    A = <* <* A*(1,1) *> , <* A*(2,1) *>, <* A*(3,1) *> *> by A6,Th27;
    then
A7: A.1 = <* A*(1,1) *> & A.2 = <* A*(2,1) *> & A.3 = <* A*(3,1) *>;
    dom M2F p = Seg 3 by A3,FINSEQ_1:def 3;
    then 1 in dom M2F p & 2 in dom M2F p & 3 in dom M2F p by FINSEQ_1:1;
    then A.1 = <* (M2F p).1 *> & A.2 = <* (M2F p).2 *> &
      A.3 = <* (M2F p).3 *> by A4,MATRIXR1:def 9;
    then A = <* <* (M2F p).1 *> , <* (M2F p).2 *>, <* (M2F p).3 *> *>
              by A6,Th27,A7
          .= F2M (M2F p) by A3,ANPROJ_8:def 1
          .= p by A1,ANPROJ_8:85;
    hence thesis;
  end;
