
theorem ass0:
for R being domRing,
    c being non zero Element of R
for b,a,d being Element of R
st a * b is_associated_to c * d & a is_associated_to c
holds b is_associated_to d
proof
let R be domRing,
    c be non zero Element of R, b,a,d be Element of R;
assume AS: a * b is_associated_to c * d & a is_associated_to c;
H0: c <> 0.R;
consider u being Element of R such that
H1: u is unital & a * b * u = c * d by AS,GCD_1:18;
consider v being Element of R such that
H2: v is unital & c * v = a by AS,GCD_1:18;
H3: c * (v * b * u) = c * (v * (b * u)) by GROUP_1:def 3
                   .= (c * v) * (b * u) by GROUP_1:def 3
                   .= c * d by H1,H2,GROUP_1:def 3;
c is right_mult-cancelable by H0,ALGSTR_0:def 37;
then d = v * b * u by H3 .= (u * v) * b by GROUP_1:def 3;
hence thesis by H1,H2,GCD_1:18;
end;
