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 Th21:
  for R being Ring, a being Element of R
  holds ((n,n)-->a)@ = (n,n)-->a
proof
    let R be Ring, a be Element of R;
    len ((n,n)-->a)=n by MATRIX_0:24;
    then
A1: len(((n,n)-->a)@)=len ((n,n)-->a) by MATRIX_0:24;
A2: Indices ((n,n)-->a)=[:Seg n, Seg n:] by MATRIX_0:24;
A3: for i,j st [i,j] in Indices (((n,n)-->a)@) holds (((n,n)-->a)@)*(i,j)=
    ((n,n)-->a)*(i,j)
    proof
      let i,j;
      assume [i,j] in Indices (((n,n)-->a)@);
      then
A4:   [i,j] in Indices (((n,n)-->a)) by MATRIX_0:26;
      then i in Seg n & j in Seg n by A2,ZFMISC_1:87;
      then [j,i] in Indices ((n,n)-->a) by A2,ZFMISC_1:87;
      then (((n,n)-->a)@)*(i,j)=((n,n)-->a)*(j,i) & ((n,n)-->a)*(j,i)=a by
MATRIX_0:46,def 6;
      hence thesis by A4,MATRIX_0:46;
    end;
    width ((n,n)-->a)=n by MATRIX_0:24;
    then width (((n,n)-->a)@) = width ((n,n)-->a) by MATRIX_0:24;
    hence thesis by A1,A3,MATRIX_0:21;
end;
