let n be Nat; :: thesis: for K being Field
for M1, M2 being Matrix of n,K st M1 is Nilpotent & M2 is Nilpotent & M1 * M2 = - (M2 * M1) & n > 0 holds
M1 + M2 is Nilpotent
let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 is Nilpotent & M2 is Nilpotent & M1 * M2 = - (M2 * M1) & n > 0 holds
M1 + M2 is Nilpotent
let M1, M2 be Matrix of n,K; :: thesis: ( M1 is Nilpotent & M2 is Nilpotent & M1 * M2 = - (M2 * M1) & n > 0 implies M1 + M2 is Nilpotent )
assume A1: ( M1 is Nilpotent & M2 is Nilpotent & M1 * M2 = - (M2 * M1) & n > 0 ) ; :: thesis: M1 + M2 is Nilpotent
then A2: ( M1 * M1 = 0. K,n & M2 * M2 = 0. K,n ) by Def2;
A3: ( len M1 = n & width M1 = n & len M2 = n & width M2 = n ) by MATRIX_1:25;
A4: ( len (M1 + M2) = n & width (M1 + M2) = n ) by MATRIX_1:25;
A5: ( len (M1 * M1) = n & width (M1 * M1) = n & len (M1 * M2) = n & width (M1 * M2) = n ) by MATRIX_1:25;
A6: ( len (M2 * M1) = n & width (M2 * M1) = n & len (M2 * M2) = n & width (M2 * M2) = n & len (- (M2 * M1)) = n & width (- (M2 * M1)) = n ) by MATRIX_1:25;
A7: ( len ((M1 * M1) + (M2 * M1)) = n & width ((M1 * M1) + (M2 * M1)) = n & len ((M1 * M2) + (M2 * M2)) = n & width ((M1 * M2) + (M2 * M2)) = n ) by MATRIX_1:25;
(M1 + M2) * (M1 + M2) = ((M1 + M2) * M1) + ((M1 + M2) * M2) by A1, A3, A4, MATRIX_4:62
.= ((M1 * M1) + (M2 * M1)) + ((M1 + M2) * M2) by A1, A3, MATRIX_4:63
.= ((M1 * M1) + (M2 * M1)) + ((M1 * M2) + (M2 * M2)) by A1, A3, MATRIX_4:63
.= (((M1 * M1) + (M2 * M1)) + (M1 * M2)) + (M2 * M2) by A5, A6, A7, MATRIX_3:5
.= ((M1 * M1) + ((M2 * M1) + (- (M2 * M1)))) + (M2 * M2) by A1, A5, A6, MATRIX_3:5
.= ((M1 * M1) + (0. K,n,n)) + (M2 * M2) by A6, MATRIX_4:2
.= (0. K,n) + (0. K,n) by A2, MATRIX_3:6
.= 0. K,n,n by MATRIX_3:6
.= 0. K,n ;
hence M1 + M2 is Nilpotent by Def2; :: thesis: verum