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

theorem Th29:
  for A being Matrix of n,D holds (A@)@=A
proof
  let A be Matrix of n,D;
  reconsider N=A@ as Matrix of n,D;
A1: len A=n & width A=n by MATRIX_0:24;
A2: Indices (N@)=[:Seg n,Seg n :] by MATRIX_0:24
    .=Indices A by MATRIX_0:24;
A3: for i,j being Nat st [i,j] in Indices (N@) holds (N@)*(i,j) = A*(i,j)
  proof
    let i,j be Nat;
    assume
A4: [i,j] in Indices (N@);
    then [j,i] in Indices N by MATRIX_0:def 6;
    then (N@)*(i,j)=N*(j,i) by MATRIX_0:def 6;
    hence thesis by A2,A4,MATRIX_0:def 6;
  end;
  len (N@)=n & width (N@)=n by MATRIX_0:24;
  hence thesis by A1,A3,MATRIX_0:21;
end;
