let L be non empty right_complementable add-associative right_zeroed unital left-distributive doubleLoopStr ; :: thesis: for z0, z1, x being Element of L holds eval <%(0. L),z1%>,x = z1 * x
let z0, z1, x be Element of L; :: thesis: eval <%(0. L),z1%>,x = z1 * x
thus eval <%(0. L),z1%>,x = (0. L) + (z1 * x) by Th45
.= z1 * x by RLVECT_1:10 ; :: thesis: verum