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 Th55:
  Col(1.(F_Real,3),1) = <* 1,0,0 *> &
  Col(1.(F_Real,3),2) = <* 0,1,0 *> &
  Col(1.(F_Real,3),3) = <* 0,0,1 *>
  proof
A1: 1.(F_Real,3) = MXF2MXR 1.(F_Real,3) by MATRIXR1:def 2
                .= 1_Rmatrix(3) by MATRIXR2:def 2;
A2: [1,1] in Indices(1.(F_Real,3)) by Th1,MATRIX_0:24;
A3: 1 in Seg 3 & 2 in Seg 3 & 3 in Seg 3 by FINSEQ_1:1;
    then 1 in dom 1_Rmatrix(3) by Th13; then
A4: Col(1_Rmatrix(3),1).1 = 1.(F_Real,3) *(1,1) by A1,MATRIX_0:def 8
                         .= 1.F_Real by A2,MATRIX_1:def 3
                         .= 1 by STRUCT_0:def 7;
A5: [2,1] in Indices(1.(F_Real,3)) by Th1,MATRIX_0:24;
    2 in dom 1_Rmatrix(3) by A3,Th13; then
A6: Col(1_Rmatrix(3),1).2 = 1.(F_Real,3) *(2,1) by A1,MATRIX_0:def 8
                         .= 0.F_Real by A5,MATRIX_1:def 3
                         .= 0 by STRUCT_0:def 6;
A7: [3,1]in Indices(1.(F_Real,3)) by Th1,MATRIX_0:24;
    3 in dom 1_Rmatrix(3) by A3,Th13; then
A8: Col(1_Rmatrix(3),1).3 = 1.(F_Real,3) *(3,1) by A1,MATRIX_0:def 8
                         .= 0.F_Real by A7,MATRIX_1:def 3
                         .= 0 by STRUCT_0:def 6;
    len Col(1_Rmatrix(3),1) = len 1.(F_Real,3) by A1,MATRIX_0:def 8
                           .= 3 by MATRIX_0:24;
    hence Col(1.(F_Real,3),1) = <* 1,0,0 *> by A1,A4,A6,A8,FINSEQ_1:45;
A9: [1,2] in Indices(1.(F_Real,3)) by Th1,MATRIX_0:24;
A10: 1 in Seg 3 & 2 in Seg 3 & 3 in Seg 3 by FINSEQ_1:1;
    then 1 in dom 1_Rmatrix(3) by Th13; then
A11: Col(1_Rmatrix(3),2).1 = 1.(F_Real,3) *(1,2) by A1,MATRIX_0:def 8
                         .= 0.F_Real by A9,MATRIX_1:def 3
                         .= 0 by STRUCT_0:def 6;
A12: [2,2] in Indices(1.(F_Real,3)) by Th1,MATRIX_0:24;
    2 in dom 1_Rmatrix(3) by A10,Th13; then
A13: Col(1_Rmatrix(3),2).2 = 1.(F_Real,3) *(2,2) by A1,MATRIX_0:def 8
                         .= 1.F_Real by A12,MATRIX_1:def 3
                         .= 1 by STRUCT_0:def 7;
A14: [3,2]in Indices(1.(F_Real,3)) by Th1,MATRIX_0:24;
    3 in dom 1_Rmatrix(3) by A10,Th13; then
A15: Col(1_Rmatrix(3),2).3 = 1.(F_Real,3) *(3,2) by A1,MATRIX_0:def 8
                         .= 0.F_Real by A14,MATRIX_1:def 3
                         .= 0 by STRUCT_0:def 6;
    len Col(1_Rmatrix(3),2) = len 1.(F_Real,3) by A1,MATRIX_0:def 8
                           .= 3 by MATRIX_0:24;
    hence Col(1.(F_Real,3),2) = <* 0,1,0 *> by A1,A11,A13,A15,FINSEQ_1:45;
A16: [1,3] in Indices(1.(F_Real,3)) by Th1,MATRIX_0:24;
A17: 1 in Seg 3 & 2 in Seg 3 & 3 in Seg 3 by FINSEQ_1:1;
    then 1 in dom 1_Rmatrix(3) by Th13; then
A18: Col(1_Rmatrix(3),3).1 = 1.(F_Real,3) *(1,3) by A1,MATRIX_0:def 8
                         .= 0.F_Real by A16,MATRIX_1:def 3
                         .= 0 by STRUCT_0:def 6;
A19: [2,3] in Indices(1.(F_Real,3)) by Th1,MATRIX_0:24;
    2 in dom 1_Rmatrix(3) by A17,Th13; then
A20: Col(1_Rmatrix(3),3).2 = 1.(F_Real,3) *(2,3) by A1,MATRIX_0:def 8
                         .= 0.F_Real by A19,MATRIX_1:def 3
                         .= 0 by STRUCT_0:def 6;
A21: [3,3]in Indices(1.(F_Real,3)) by Th1,MATRIX_0:24;
    3 in dom 1_Rmatrix(3) by A17,Th13; then
A22: Col(1_Rmatrix(3),3).3 = 1.(F_Real,3) *(3,3) by A1,MATRIX_0:def 8
                         .= 1.F_Real by A21,MATRIX_1:def 3
                         .= 1 by STRUCT_0:def 7;
    len Col(1_Rmatrix(3),3) = len 1.(F_Real,3) by A1,MATRIX_0:def 8
                           .= 3 by MATRIX_0:24;
    hence Col(1.(F_Real,3),3) = <* 0,0,1 *> by A1,A18,A20,A22,FINSEQ_1:45;
  end;
