theorem Th68:
  Q is struct-invariant & v is y-omitting implies
  (canonical_homomorphism Q).v is y-omitting
  proof assume
AA: Q is struct-invariant;
    assume Z0: v is y-omitting;
    set H = canonical_homomorphism Q;
    assume Z1: H.v is non y-omitting;
    reconsider Hv = H.v as Element of Free(S,Y) by MSAFREE4:39;
    consider x being Element of Y.s such that
A1: for v1 st v1 in {v,Hv} holds v1 is x-omitting by Th3A;
    v in {v,Hv} & Hv in {v,Hv} by TARSKI:def 2;
    then
A2: v is x-omitting & Hv is x-omitting by A1;
    then
A4: x <> y & H.v is x-omitting by Z1;
A3: Hom(Q,x,y).(H.v) = (Hom(Q,x,y)**H).v by Th14
    .= (H**Hom(Free(S,Y),x,y)).v by Th66
    .= H.(Hom(Free(S,Y),x,y).v) by Th14
    .= H.v by Z0,A2,Th78;
    Hom(Q,x,y) = Hom(Q,y,x) by Th56;
    hence thesis by Z1,A3,AA,A4,Th79;
  end;
