let n be Nat; :: thesis: for K being Field
for M1 being Matrix of n,K st M1 is invertible & M1 is Idempotent holds
M1 ~ is Idempotent
let K be Field; :: thesis: for M1 being Matrix of n,K st M1 is invertible & M1 is Idempotent holds
M1 ~ is Idempotent
let M1 be Matrix of n,K; :: thesis: ( M1 is invertible & M1 is Idempotent implies M1 ~ is Idempotent )
assume A1: ( M1 is invertible & M1 is Idempotent ) ; :: thesis: M1 ~ is Idempotent
then M1 ~ = (M1 * M1) ~ by Def1
.= (M1 ~ ) * (M1 ~ ) by A1, MATRIX_6:37 ;
hence M1 ~ is Idempotent by Def1; :: thesis: verum