let n be Nat; :: thesis: for K being Field

for M1, M2 being Matrix of n,K st n > 0 & M1 is Nilpotent & M2 is Nilpotent & M1 commutes_with M2 & M1 * M2 = 0. (K,n) holds

M1 + M2 is Nilpotent

let K be Field; :: thesis: for M1, M2 being Matrix of n,K st n > 0 & M1 is Nilpotent & M2 is Nilpotent & M1 commutes_with M2 & M1 * M2 = 0. (K,n) holds

M1 + M2 is Nilpotent

let M1, M2 be Matrix of n,K; :: thesis: ( n > 0 & M1 is Nilpotent & M2 is Nilpotent & M1 commutes_with M2 & M1 * M2 = 0. (K,n) implies M1 + M2 is Nilpotent )

assume that

A1: n > 0 and

A2: ( M1 is Nilpotent & M2 is Nilpotent ) and

A3: M1 commutes_with M2 and

A4: M1 * M2 = 0. (K,n) ; :: thesis: M1 + M2 is Nilpotent

A5: M1 * M2 = 0. (K,n,n) by A4;

A6: ( M1 * M1 = 0. (K,n) & M2 * M2 = 0. (K,n) ) by A2;

(M1 + M2) * (M1 + M2) = (((M1 * M1) + (0. (K,n))) + (0. (K,n))) + (M2 * M2) by A1, A3, A4, MATRIX_6:35

.= ((M1 * M1) + (0. (K,n))) + (M2 * M2) by A5, MATRIX_3:4

.= (0. (K,n)) + (0. (K,n)) by A6, A5, MATRIX_3:4

.= 0. (K,n,n) by MATRIX_3:4

.= 0. (K,n) ;

hence M1 + M2 is Nilpotent ; :: thesis: verum

for M1, M2 being Matrix of n,K st n > 0 & M1 is Nilpotent & M2 is Nilpotent & M1 commutes_with M2 & M1 * M2 = 0. (K,n) holds

M1 + M2 is Nilpotent

let K be Field; :: thesis: for M1, M2 being Matrix of n,K st n > 0 & M1 is Nilpotent & M2 is Nilpotent & M1 commutes_with M2 & M1 * M2 = 0. (K,n) holds

M1 + M2 is Nilpotent

let M1, M2 be Matrix of n,K; :: thesis: ( n > 0 & M1 is Nilpotent & M2 is Nilpotent & M1 commutes_with M2 & M1 * M2 = 0. (K,n) implies M1 + M2 is Nilpotent )

assume that

A1: n > 0 and

A2: ( M1 is Nilpotent & M2 is Nilpotent ) and

A3: M1 commutes_with M2 and

A4: M1 * M2 = 0. (K,n) ; :: thesis: M1 + M2 is Nilpotent

A5: M1 * M2 = 0. (K,n,n) by A4;

A6: ( M1 * M1 = 0. (K,n) & M2 * M2 = 0. (K,n) ) by A2;

(M1 + M2) * (M1 + M2) = (((M1 * M1) + (0. (K,n))) + (0. (K,n))) + (M2 * M2) by A1, A3, A4, MATRIX_6:35

.= ((M1 * M1) + (0. (K,n))) + (M2 * M2) by A5, MATRIX_3:4

.= (0. (K,n)) + (0. (K,n)) by A6, A5, MATRIX_3:4

.= 0. (K,n,n) by MATRIX_3:4

.= 0. (K,n) ;

hence M1 + M2 is Nilpotent ; :: thesis: verum