
theorem Th29:
  for T being non empty non void reflexive transitive TA-structure
for t being type of T, a be adjective of T holds apply(<*a*>, t) = <*t, a ast t
  *>
proof
  let T be non empty non void reflexive transitive TA-structure;
  let t be type of T, a be adjective of T;
A1: <*a*>.1 = a;
A2: apply(<*a*>, t).1 = t by Def19;
A3: len <*a*> = 1 by FINSEQ_1:40;
  then
A4: len apply(<*a*>, t) = 1+1 by Def19;
  1 in dom <*a*> by A3,FINSEQ_3:25;
  then apply(<*a*>, t).(len <*a*>+1) = a ast t by A3,A2,A1,Def19;
  hence thesis by A3,A4,A2,FINSEQ_1:44;
end;
