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;

theorem Th7:
  B is associative implies product(B,n) is associative
proof
  assume
A1: B is associative;
  now
    let x,y,z be Element of n-tuples_on D;
    thus product(B,n).(product(B,n).(x,y),z) = product(B,n).(B.:(x,y),z)
      by Def1
      .= B.:(B.:(x,y),z) by Def1
      .= B.:(x,B.:(y,z)) by A1,FINSEQOP:28
      .= product(B,n).(x,B.:(y,z)) by Def1
      .= product(B,n).(x,product(B,n).(y,z)) by Def1;
  end;
  hence thesis;
end;
