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 Th15:
  for u being UnOps of a for f being Element of product a, i being
  Element of dom a holds (Frege u).f.i = (u.i).(f.i)
proof
  let u be UnOps of a, f be Element of product a;
  let i be Element of dom a;
A1: dom u = Seg len u & doms u = a by Th14,FINSEQ_1:def 3;
  dom a = Seg len a & len a = len u by Th12,FINSEQ_1:def 3;
  hence thesis by A1,FUNCT_6:37;
end;
