let L be non empty right_complementable distributive Abelian add-associative right_zeroed doubleLoopStr ; :: thesis: for a, b being Element of L
for p being sequence of L holds (a + b) * p = (a * p) + (b * p)
let a, b be Element of L; :: thesis: for p being sequence of L holds (a + b) * p = (a * p) + (b * p)
let p be sequence of L; :: thesis: (a + b) * p = (a * p) + (b * p)
for i being Element of NAT holds ((a + b) * p) . i = ((a * p) + (b * p)) . i

proof

let i be Element of NAT ; :: thesis: ((a + b) * p) . i = ((a * p) + (b * p)) . i
thus ((a + b) * p) . i = (a + b) * (p . i) by POLYNOM5:def 4
.= (a * (p . i)) + (b * (p . i)) by VECTSP_1:def 7
.= ((a * p) . i) + (b * (p . i)) by POLYNOM5:def 4
.= ((a * p) . i) + ((b * p) . i) by POLYNOM5:def 4
.= ((a * p) + (b * p)) . i by NORMSP_1:def 2 ; :: thesis: verum

end;

hence (a + b) * p = (a * p) + (b * p) by FUNCT_2:63; :: thesis: verum