reserve G for Abelian add-associative right_complementable right_zeroed
  non empty addLoopStr;
reserve GS for non empty addLoopStr;
reserve F for Field;
reserve F for Field,
  n for Nat,
  D for non empty set,
  d for Element of D,
  B for BinOp of D,
  C for UnOp of D;
reserve x,y for set;
reserve D for non empty set,
  H,G for BinOp of D,
  d for Element of D,
  t1,t2 for Element of n-tuples_on D;
reserve x,y,z for set,
  A for AbGroup;
reserve a for Domain-Sequence,
  i for Element of dom a,
  p for FinSequence;

theorem Th14:
  for u being UnOps of a holds doms u = a & product rngs u c= product a
proof
  let u be UnOps of a;
A2: dom a = Seg len a & dom u = Seg len u by FINSEQ_1:def 3;
A3: len u = len a by Th12;
A4: dom doms u = dom u by FUNCT_6:def 2;
A5: now
    let x be object;
    assume x in dom u;
    then reconsider i = x as Element of dom a by A2,Th12;
    (rngs u).i = rng (u.i) by A2,A3,FUNCT_6:22;
    hence (rngs u).x c= a.x by RELAT_1:def 19;
  end;
  now
    let x be object;
    assume x in dom u;
    then reconsider i = x as Element of dom a by A2,Th12;
    (doms u).i = dom (u.i) by A2,A3,FUNCT_6:22
      .= a.i by FUNCT_2:def 1;
    hence (doms u).x = a.x;
  end;
  hence doms u = a by A2,Th12,A4;
  dom rngs u = dom u by FUNCT_6:def 3;
  hence thesis by A2,A3,A5,CARD_3:27;
end;
