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 commutes_with M2 holds
M1 * M1 commutes_with M2 * M2
let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 is Idempotent & M2 is Idempotent & M1 commutes_with M2 holds
M1 * M1 commutes_with M2 * M2
let M1, M2 be Matrix of n,K; :: thesis: ( M1 is Idempotent & M2 is Idempotent & M1 commutes_with M2 implies M1 * M1 commutes_with M2 * M2 )
assume that
A1: M1 is Idempotent and
A2: ( M2 is Idempotent & M1 commutes_with M2 ) ; :: thesis: M1 * M1 commutes_with M2 * M2
M1 * M1 = M1 by A1, Def1;
hence M1 * M1 commutes_with M2 * M2 by A2, Def1; :: thesis: verum