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

theorem Th63:
  (for i st [i,i] in Indices 1_Rmatrix n holds (1_Rmatrix n)*(i,i)
= 1) & for i,j st [i,j] in Indices (1_Rmatrix n) & i <> j holds (1_Rmatrix n)*(
  i,j) = 0
proof
  set K=F_Real;
  thus for i st [i,i] in Indices 1_Rmatrix n holds (1_Rmatrix n)*(i,i) = 1
  proof
    let i;
    assume [i,i] in Indices 1_Rmatrix n;
    hence (1_Rmatrix n)*(i,i) = 1_ F_Real by MATRIX_1:def 3
      .=1;
  end;
  let i,j;
  assume [i,j] in Indices 1_Rmatrix n & i <> j;
  hence (1_Rmatrix n)*(i,j) = 0.K by MATRIX_1:def 3
    .=0;
end;
