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

theorem Th10:
  M1 is invertible Idempotent implies M1=1.(K,n)
proof
A1: len M1=n & width M1=n by MATRIX_0:24;
A2: width (M1~)=n by MATRIX_0:24;
  assume
A3: M1 is invertible Idempotent;
  then
A4: M1~ is_reverse_of M1 by MATRIX_6:def 4;
  M1*M1=M1 by A3;
  then 1.(K,n)=M1~*(M1*M1) by A4,MATRIX_6:def 2
    .=(M1~*M1)*M1 by A1,A2,MATRIX_3:33
    .=(1.(K,n))*M1 by A4,MATRIX_6:def 2
    .=M1 by MATRIX_3:18;
  hence thesis;
end;
