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

theorem
  M1 is Idempotent & M2 is Idempotent & M1*M2=-(M2*M1) implies M1+M2 is
  Idempotent
proof
  assume that
A1: M1 is Idempotent & M2 is Idempotent and
A2: M1*M2=-(M2*M1);
A3: M1*M1=M1 & M2*M2=M2 by A1;
A5: len (M1*M2)=n & width (M1*M2)=n by MATRIX_0:24;
A6: len M2=n & width M2=n by MATRIX_0:24;
A7: len (M1*M1+M2*M1)=n & width (M1*M1+M2*M1)=n by MATRIX_0:24;
A8: len (M2*M1)=n & width (M2*M1)=n by MATRIX_0:24;
A9: len (M1*M1)=n & width (M1*M1)=n by MATRIX_0:24;
A10: len M1=n & width M1=n by MATRIX_0:24;
    len (M1+M2)=n & width (M1+M2)=n by MATRIX_0:24;
    then (M1+M2)*(M1+M2)=(M1+M2)*M1+(M1+M2)*M2 by A10,A6,MATRIX_4:62
      .=M1*M1+M2*M1+(M1+M2)*M2 by A10,A6,MATRIX_4:63
      .=M1*M1+M2*M1+(M1*M2+M2*M2) by A10,A6,MATRIX_4:63
      .=M1*M1+M2*M1+M1*M2+M2*M2 by A5,A7,MATRIX_3:3
      .=(M1*M1+(M2*M1+(-(M2*M1))))+M2*M2 by A2,A9,A8,MATRIX_3:3
      .=M1*M1+0.(K,n,n)+M2*M2 by A8,MATRIX_4:2
      .=M1+M2 by A3,MATRIX_3:4;
    hence thesis;
end;
