reserve x for set,
  D for non empty set,
  k,n,m,i,j,l for Nat,
  K for Field;

theorem Th64:
  (1_Rmatrix(n))@=1_Rmatrix(n)
proof
  per cases;
  suppose
A1: n>0;
A2: len (1_Rmatrix(n))=n by MATRIX_0:24;
A3: Indices (1_Rmatrix(n)) = [:Seg n, Seg n:] by MATRIX_0:24;
A4: for i,j being Nat st [i,j] in Indices (1_Rmatrix(n)) holds (1_Rmatrix(
    n))*(i,j)=((1_Rmatrix(n))@)*(i,j)
    proof
      let i,j be Nat;
      reconsider i0=i,j0=j as Nat;
      assume
A5:   [i,j] in Indices (1_Rmatrix(n));
      then i in Seg n & j in Seg n by A3,ZFMISC_1:87;
      then
A6:   [j,i] in Indices (1_Rmatrix(n) ) by A3,ZFMISC_1:87;
      per cases;
      suppose
        i=j;
        hence thesis by A5,MATRIX_0:def 6;
      end;
      suppose
        i<>j;
        then (1_Rmatrix(n))*(i0,j0)=0 & (1_Rmatrix(n))*(j0,i0)=0 by A5,A6,Th63;
        hence thesis by A6,MATRIX_0:def 6;
      end;
    end;
A7: width (1_Rmatrix(n))=n by MATRIX_0:24;
    then
    len ((1_Rmatrix(n))@)=width (1_Rmatrix(n)) & width ((1_Rmatrix(n))@)=
    len ( 1_Rmatrix n) by A1,MATRIX_0:54;
    hence thesis by A7,A2,A4,MATRIX_0:21;
  end;
  suppose
    n=0;
    hence thesis by MATRIX_0:45;
  end;
end;
