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 Th16:
  for d,d9 being BinOp of product a st
  for f,g being Element of product a, i being Element of dom a holds
    (d.(f,g)).i = (d9.(f,g)).i holds d = d9
proof
  let d,d9 be BinOp of product a such that
A1: for f,g be Element of product a, i be Element of dom a holds
    (d.(f,g)).i = (d9.(f,g)).i;
  now
    let f,g be Element of product a;
    dom (d.(f,g)) = dom a & dom (d9.(f,g)) = dom a by CARD_3:9;
    hence d.(f,g) = d9.(f,g) by A1;
  end;
  hence thesis;
end;
