let n be set ; :: thesis: for L being non empty right_complementable add-associative right_zeroed addLoopStr
for p being Series of n,L holds
( (- p) + p = 0_ n,L & p + (- p) = 0_ n,L )
let L be non empty right_complementable add-associative right_zeroed addLoopStr ; :: thesis: for p being Series of n,L holds
( (- p) + p = 0_ n,L & p + (- p) = 0_ n,L )
let p be Series of n,L; :: thesis: ( (- p) + p = 0_ n,L & p + (- p) = 0_ n,L )
set q = (- p) + p;

now

let b be Element of Bags n; :: thesis: ((- p) + p) . b = (0_ n,L) . b
thus ((- p) + p) . b = ((- p) . b) + (p . b) by POLYNOM1:def 21
.= (- (p . b)) + (p . b) by POLYNOM1:def 22
.= 0. L by RLVECT_1:16
.= (0_ n,L) . b by POLYNOM1:81 ; :: thesis: verum

end;

hence (- p) + p = 0_ n,L by FUNCT_2:113; :: thesis: p + (- p) = 0_ n,L
set q = p + (- p);

now

let b be Element of Bags n; :: thesis: (p + (- p)) . b = (0_ n,L) . b
thus (p + (- p)) . b = (p . b) + ((- p) . b) by POLYNOM1:def 21
.= (p . b) + (- (p . b)) by POLYNOM1:def 22
.= 0. L by RLVECT_1:16
.= (0_ n,L) . b by POLYNOM1:81 ; :: thesis: verum

end;

hence p + (- p) = 0_ n,L by FUNCT_2:113; :: thesis: verum