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 Th51:
  for pr being FinSequence of D st len pr = 3 holds
  Col(<*pr*>,1) = <* pr.1 *> &
  Col(<*pr*>,2) = <* pr.2 *> & Col(<*pr*>,3) = <* pr.3 *>
  proof
    let pr be FinSequence of D;
    assume len pr = 3;
    then
A2: Indices <*pr*> = [:Seg 1,Seg 3:] by MATRIX_0:23;
    consider p be FinSequence of D such that
A3: p = <*pr*>.1 and
A4: (<*pr*>)*(1,1) = p.1 by A2,Th3,MATRIX_0:def 5;
A5: len Col(<*pr*>,1) = len <*pr*> by MATRIX_0:def 8;
A6: len <*pr*> = 1 by FINSEQ_1:39;
    then
A7: dom <*pr*> = Seg 1 by FINSEQ_1:def 3;
    then Col(<*pr*>,1).1 = <*pr*>*(1,1) by FINSEQ_1:1,MATRIX_0:def 8
                        .= pr.1 by A4,A3;
    hence Col(<*pr*>,1) = <* pr.1 *>  by A6,A5,FINSEQ_1:40;
    consider p be FinSequence of D such that
A8: p = <*pr*>.1 and
A9: (<*pr*>)*(1,2) = p.2 by A2,Th3,MATRIX_0:def 5;
A10: len Col(<*pr*>,2) = len <*pr*> by MATRIX_0:def 8;
    Col(<*pr*>,2).1 = <*pr*>*(1,2) by A7,FINSEQ_1:1,MATRIX_0:def 8
                   .= pr.2 by A9,A8;
    hence Col(<*pr*>,2) = <* pr.2 *>  by A6,A10,FINSEQ_1:40;
    consider p be FinSequence of D such that
A11: p = <*pr*>.1 and
A12: (<*pr*>)*(1,3) = p.3 by A2,Th3,MATRIX_0:def 5;
A13: len Col(<*pr*>,3) = len <*pr*> by MATRIX_0:def 8;
    Col(<*pr*>,3).1 = <*pr*>*(1,3) by A7,FINSEQ_1:1,MATRIX_0:def 8
                   .= pr.3 by A12,A11;
    hence Col(<*pr*>,3) = <* pr.3 *>  by A6,A13,FINSEQ_1:40;
  end;
