reserve i,n for Nat,
  K for Field,
  M1,M2,M3,M4 for Matrix of n,K;

theorem
  n>0 & M1 is Nilpotent implies not M1 is invertible
proof
  assume that
A1: n>0 and
A2: M1 is Nilpotent;
A3: len M1=n & width M1=n by MATRIX_0:24;
  assume M1 is invertible;
  then consider M2 being Matrix of n,K such that
A4: M1 is_reverse_of M2 by MATRIX_6:def 3;
A5: width M2=n by MATRIX_0:24;
A6: len M2=n by MATRIX_0:24;
  M1=M1*(1.(K,n)) by MATRIX_3:19
    .=M1*(M1*M2) by A4,MATRIX_6:def 2
    .=(M1*M1)*M2 by A3,A6,MATRIX_3:33
    .=(0.(K,n))*M2 by A2
    .=0.(K,n,n) by A6,A5,MATRIX_6:1;
  then
A7: M1*M2=0.(K,n) by A6,A5,MATRIX_6:1;
  M1*M2= 1.(K,n) by A4,MATRIX_6:def 2;
  hence contradiction by A1,A7,Th28;
end;
