let n be Nat; :: thesis: 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; :: thesis: 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; :: thesis: ( 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 ; :: thesis: M1 * M2 is Idempotent
M1 = 1. (K,n) by A1, A3, Th10;
hence M1 * M2 is Idempotent by A2, MATRIX_3:18; :: thesis: verum