
theorem Th44:
  for T being Noetherian adj-structured reflexive transitive
  antisymmetric with_suprema non void TA-structure for t being type of T for v
being FinSequence of the adjectives of T st v is_applicable_to t holds rng v c=
  adjs (v ast t)
proof
  let T be Noetherian adj-structured reflexive transitive antisymmetric
  with_suprema non void TA-structure;
  let t be type of T;
  let v be FinSequence of the adjectives of T such that
A1: v is_applicable_to t;
  let a be object;
  assume
A2: a in rng v;
  then consider x being object such that
A3: x in dom v and
A4: a = v.x by FUNCT_1:def 3;
  reconsider a as adjective of T by A2;
  reconsider x as Element of NAT by A3;
A5: x >= 1 by A3,FINSEQ_3:25;
  then
A6: x+1 >= 1 by NAT_1:13;
A7: len apply(v,t) = len v+1 by Def19;
A8: x <= len v by A3,FINSEQ_3:25;
  then x+1 <= len apply(v,t) by A7,XREAL_1:6;
  then x+1 in dom apply(v,t) by A6,FINSEQ_3:25;
  then
A9: apply(v,t).(x+1) in rng apply(v,t) by FUNCT_1:3;
  x < len apply(v,t) by A8,A7,NAT_1:13;
  then x in dom apply(v,t) by A5,FINSEQ_3:25;
  then apply(v,t).x in rng apply(v,t) by FUNCT_1:3;
  then reconsider s = apply(v,t).x as type of T;
  a ast s = apply(v,t).(x+1) by A3,A4,Def19;
  then a ast s >= v ast t by A1,A9,Th42;
  then
A10: adjs (a ast s) c= adjs(v ast t) by Th10;
  a is_applicable_to s by A1,A3,A4;
  then a in adjs (a ast s) by Th21;
  hence thesis by A10;
end;
