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 Th57:
  <*<e1>*>@ = <* <* 1 *>, <* 0 *> , <* 0 *> *> &
  <*<e2>*>@ = <* <* 0 *>, <* 1 *> , <* 0 *> *> &
  <*<e3>*>@ = <* <* 0 *>, <* 0 *> , <* 1 *> *>
  proof
    <e1> in REAL 3; then
A1: <e1> in 3-tuples_on REAL by EUCLID:def 1;
A2: len <*<e1>*> = 1 by FINSEQ_1:39;
    rng <*<e1>*> = {<e1>} by FINSEQ_1:39;
    then <e1> in rng <*<e1>*> by TARSKI:def 1; then
A4: width <*<e1>*> = len <e1> by A2,MATRIX_0:def 3
                      .= 3 by A1,FINSEQ_2:133;
A5: width (<*<e1>*>@) = len <*<e1>*> by A4,MATRIX_0:29
                       .= 1 by FINSEQ_1:39;
    now
      thus len (<*<e1>*>@) = 3 by MATRIX_0:def 6,A4; then
A7:   <*<e1>*>@ is Matrix of 3,1,F_Real by A5,MATRIX_0:20;
A8:   1 in Seg 3 by FINSEQ_1:1;
      hence <*<e1>*>@.1 = Line (<*<e1>*>@,1) by A7,MATRIX_0:52
                       .= <* 1 *> by A8,A4,MATRIX_0:59,Th52;
A9:   2 in Seg 3 by FINSEQ_1:1;
      hence <*<e1>*>@.2 = Line (<*<e1>*>@,2) by A7,MATRIX_0:52
                       .= <* 0 *> by A9,A4,MATRIX_0:59,Th52;
A10:  3 in Seg 3 by FINSEQ_1:1;
      hence <*<e1>*>@.3 = Line (<*<e1>*>@,3) by A7,MATRIX_0:52
                       .= <* 0 *> by A10,A4,MATRIX_0:59,Th52;
    end;
    hence <*<e1>*>@ = <* <* 1 *>, <* 0 *> , <* 0 *> *> by FINSEQ_1:45;
    <e2> in REAL 3; then
A11: <e2> in 3-tuples_on REAL by EUCLID:def 1;
A12: len <*<e2>*> = 1 by FINSEQ_1:39;
    rng <*<e2>*> = {<e2>} by FINSEQ_1:39;
    then <e2> in rng <*<e2>*> by TARSKI:def 1; then
A13bis: width <*<e2>*> = len <e2> by A12,MATRIX_0:def 3
                      .= 3 by A11,FINSEQ_2:133; then
A14: width (<*<e2>*>@) = len <*<e2>*> by MATRIX_0:29
                       .= 1 by FINSEQ_1:39;
    now
      thus len (<*<e2>*>@) = 3 by MATRIX_0:def 6,A13bis; then
A16:  <*<e2>*>@ is Matrix of 3,1,F_Real by A14,MATRIX_0:20;
A17:  1 in Seg 3 by FINSEQ_1:1;
      hence <*<e2>*>@.1 = Line (<*<e2>*>@,1) by A16,MATRIX_0:52
                       .= <* 0 *> by A17,A13bis,MATRIX_0:59,Th53;
A18:  2 in Seg 3 by FINSEQ_1:1;
      hence <*<e2>*>@.2 = Line (<*<e2>*>@,2) by A16,MATRIX_0:52
                       .= <* 1 *> by A18,A13bis,MATRIX_0:59,Th53;
A19:  3 in Seg 3 by FINSEQ_1:1;
      hence <*<e2>*>@.3 = Line (<*<e2>*>@,3) by A16,MATRIX_0:52
                       .= <* 0 *> by A19,A13bis,MATRIX_0:59,Th53;
    end;
    hence <*<e2>*>@ = <* <* 0 *>, <* 1 *> , <* 0 *> *> by FINSEQ_1:45;
    <e3> in REAL 3; then
A20: <e3> in 3-tuples_on REAL by EUCLID:def 1;
A21: len <*<e3>*> = 1 by FINSEQ_1:39;
    rng <*<e3>*> = {<e3>} by FINSEQ_1:39;
    then <e3> in rng <*<e3>*> by TARSKI:def 1; then
A23: width <*<e3>*> = len <e3> by A21,MATRIX_0:def 3
                      .= 3 by A20,FINSEQ_2:133;
A24: width (<*<e3>*>@) = len <*<e3>*> by A23,MATRIX_0:29
                      .= 1 by FINSEQ_1:39;
    now
      thus
A25:  len (<*<e3>*>@) = 3 by MATRIX_0:def 6,A23;
A26:  <*<e3>*>@ is Matrix of 3,1,F_Real by A25,A24,MATRIX_0:20;
A27:  1 in Seg 3 by FINSEQ_1:1;
      hence <*<e3>*>@.1 = Line (<*<e3>*>@,1) by A26,MATRIX_0:52
                       .= <* 0 *> by A27,A23,MATRIX_0:59,Th54;
A28:  2 in Seg 3 by FINSEQ_1:1;
      hence <*<e3>*>@.2 = Line (<*<e3>*>@,2) by A26,MATRIX_0:52
                       .= <* 0 *> by A28,A23,MATRIX_0:59,Th54;
A29:  3 in Seg 3 by FINSEQ_1:1;
      hence <*<e3>*>@.3 = Line (<*<e3>*>@,3) by A26,MATRIX_0:52
                       .= <* 1 *> by A29,A23,MATRIX_0:59,Th54;
    end;
    hence <*<e3>*>@ = <* <* 0 *>, <* 0 *> , <* 1 *> *> by FINSEQ_1:45;

  end;
