let n be Nat; for K being Field
for M1, M2 being Matrix of n,K st M1 is Idempotent & M2 is Idempotent & M1 is invertible holds
M1 * M2 is Idempotent
let K be Field; for M1, M2 being Matrix of n,K st M1 is Idempotent & M2 is Idempotent & M1 is invertible holds
M1 * M2 is Idempotent
let M1, M2 be Matrix of n,K; ( M1 is Idempotent & M2 is Idempotent & M1 is invertible implies M1 * M2 is Idempotent )
assume that
A1:
M1 is Idempotent
and
A2:
M2 is Idempotent
and
A3:
M1 is invertible
; M1 * M2 is Idempotent
M1 = 1. (K,n)
by A1, A3, Th10;
hence
M1 * M2 is Idempotent
by A2, MATRIX_3:18; verum