
theorem MLT2:
  for n being Nat holds 1.(INT.Ring,n) = 1.(F_Real,n)
  proof
    let n be Nat;
    set M = 1.(INT.Ring,n);
    set L = 1.(F_Real,n);
    P1: len M = n & width M = n & Indices M = [:Seg n, Seg n:] by MATRIX_0:24;
    P2: len L = n & width L = n & Indices L = [:Seg n, Seg n:] by MATRIX_0:24;
    for i, j being Nat st [i,j] in Indices M holds M*(i,j) = L*(i,j)
    proof
      let i, j be Nat;
      assume AS: [i,j] in Indices M;
      per cases;
      suppose Q1: i = j;
        hence M*(i,j) = 1.INT.Ring by MATRIX_1:def 3,AS
        .= 1.F_Real
        .= L*(i,j) by Q1,MATRIX_1:def 3,P1,P2,AS;
      end;
      suppose Q2: i <> j;
        hence M*(i,j) = 0.INT.Ring by MATRIX_1:def 3,AS
        .= 0.F_Real
        .= L*(i,j) by Q2,MATRIX_1:def 3,P1,P2,AS;
      end;
    end;
    hence thesis by P1,P2,ZMATRLIN:4;
  end;
