reserve i,j,n for Nat,
  K for Field,
  a for Element of K,
  M,M1,M2,M3,M4 for Matrix of n,K;
reserve A for Matrix of K;

theorem Th11:
  for R being Ring
  holds (1.(R,n))@ = 1.(R,n)
proof
  let R be Ring;
  per cases by NAT_1:3;
  suppose
A1: n>0;
A2: len (1.(R,n))=n by MATRIX_0:24;
A3: Indices (1.(R,n)) = [:Seg n, Seg n:] by MATRIX_0:24;
A4: for i,j st [i,j] in Indices (1.(R,n)) holds (1.(R,n))*(i,j)=((1.(R,n))
    @)*(i,j)
    proof
      let i,j;
      assume
A5:   [i,j] in Indices (1.(R,n));
      then i in Seg n & j in Seg n by A3,ZFMISC_1:87;
      then
A6:   [j,i] in Indices (1.(R,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.(R,n))*(i,j)=0.R & (1.(R,n))*(j,i)=0.R by A5,A6,MATRIX_1:def 3;
        hence thesis by A6,MATRIX_0:def 6;
      end;
    end;
A7: width (1.(R,n))=n by MATRIX_0:24;
    then len ((1.(R,n))@)=width (1.(R,n)) & width ((1.(R,n))@)=len (1.(R,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;
