Article XREAL_0, MML version 4.99.1005

:: XREAL_0:attrnot 1 => XREAL_0:attr 1
definition
  let a1 be set;
  attr a1 is real means
    a1 in REAL;
end;

:: XREAL_0:dfs 1
definiens
  let a1 be set;
To prove
     a1 is real
it is sufficient to prove
  thus a1 in REAL;

:: XREAL_0:def 1
theorem
for b1 being set holds
      b1 is real
   iff
      b1 in REAL;

:: XREAL_0:funcreg 1
registration
  cluster -infty -> non real;
end;

:: XREAL_0:funcreg 2
registration
  cluster +infty -> non real;
end;

:: XREAL_0:condreg 1
registration
  cluster natural -> real (set);
end;

:: XREAL_0:condreg 2
registration
  cluster real -> complex (set);
end;

:: XREAL_0:exreg 1
registration
  cluster real set;
end;

:: XREAL_0:condreg 3
registration
  cluster real -> ext-real (set);
end;

:: XREAL_0:funcreg 3
registration
  let a1 be real set;
  cluster - a1 -> complex real;
end;

:: XREAL_0:funcreg 4
registration
  let a1 be real set;
  cluster a1 " -> complex real;
end;

:: XREAL_0:funcreg 5
registration
  let a1, a2 be real set;
  cluster a1 + a2 -> real;
end;

:: XREAL_0:funcreg 6
registration
  let a1, a2 be real set;
  cluster a1 * a2 -> real;
end;

:: XREAL_0:funcreg 7
registration
  let a1, a2 be real set;
  cluster a1 - a2 -> real;
end;

:: XREAL_0:funcreg 8
registration
  let a1, a2 be real set;
  cluster a1 / a2 -> real;
end;

:: XREAL_0:exreg 2
registration
  cluster complex ext-real positive real set;
end;

:: XREAL_0:exreg 3
registration
  cluster complex ext-real negative real set;
end;

:: XREAL_0:exreg 4
registration
  cluster empty complex ext-real real set;
end;

:: XREAL_0:funcreg 9
registration
  let a1, a2 be non negative real set;
  cluster a1 + a2 -> non negative;
end;

:: XREAL_0:funcreg 10
registration
  let a1, a2 be non positive real set;
  cluster a1 + a2 -> non positive;
end;

:: XREAL_0:funcreg 11
registration
  let a1 be positive real set;
  let a2 be non negative real set;
  cluster a1 + a2 -> positive;
end;

:: XREAL_0:funcreg 12
registration
  let a1 be positive real set;
  let a2 be non negative real set;
  cluster a2 + a1 -> positive;
end;

:: XREAL_0:funcreg 13
registration
  let a1 be negative real set;
  let a2 be non positive real set;
  cluster a1 + a2 -> negative;
end;

:: XREAL_0:funcreg 14
registration
  let a1 be negative real set;
  let a2 be non positive real set;
  cluster a2 + a1 -> negative;
end;

:: XREAL_0:funcreg 15
registration
  let a1 be non positive real set;
  cluster - a1 -> complex non negative;
end;

:: XREAL_0:funcreg 16
registration
  let a1 be non negative real set;
  cluster - a1 -> complex non positive;
end;

:: XREAL_0:funcreg 17
registration
  let a1 be non negative real set;
  let a2 be non positive real set;
  cluster a1 - a2 -> non negative;
end;

:: XREAL_0:funcreg 18
registration
  let a1 be non negative real set;
  let a2 be non positive real set;
  cluster a2 - a1 -> non positive;
end;

:: XREAL_0:funcreg 19
registration
  let a1 be positive real set;
  let a2 be non positive real set;
  cluster a1 - a2 -> positive;
end;

:: XREAL_0:funcreg 20
registration
  let a1 be positive real set;
  let a2 be non positive real set;
  cluster a2 - a1 -> negative;
end;

:: XREAL_0:funcreg 21
registration
  let a1 be negative real set;
  let a2 be non negative real set;
  cluster a1 - a2 -> negative;
end;

:: XREAL_0:funcreg 22
registration
  let a1 be negative real set;
  let a2 be non negative real set;
  cluster a2 - a1 -> positive;
end;

:: XREAL_0:funcreg 23
registration
  let a1 be non positive real set;
  let a2 be non negative real set;
  cluster a1 * a2 -> non positive;
end;

:: XREAL_0:funcreg 24
registration
  let a1 be non positive real set;
  let a2 be non negative real set;
  cluster a2 * a1 -> non positive;
end;

:: XREAL_0:funcreg 25
registration
  let a1, a2 be non positive real set;
  cluster a1 * a2 -> non negative;
end;

:: XREAL_0:funcreg 26
registration
  let a1, a2 be non negative real set;
  cluster a1 * a2 -> non negative;
end;

:: XREAL_0:funcreg 27
registration
  let a1 be non positive real set;
  cluster a1 " -> complex non positive;
end;

:: XREAL_0:funcreg 28
registration
  let a1 be non negative real set;
  cluster a1 " -> complex non negative;
end;

:: XREAL_0:funcreg 29
registration
  let a1 be non negative real set;
  let a2 be non positive real set;
  cluster a1 / a2 -> non positive;
end;

:: XREAL_0:funcreg 30
registration
  let a1 be non negative real set;
  let a2 be non positive real set;
  cluster a2 / a1 -> non positive;
end;

:: XREAL_0:funcreg 31
registration
  let a1, a2 be non negative real set;
  cluster a1 / a2 -> non negative;
end;

:: XREAL_0:funcreg 32
registration
  let a1, a2 be non positive real set;
  cluster a1 / a2 -> non negative;
end;

:: XREAL_0:funcreg 33
registration
  let a1, a2 be real set;
  cluster min(a1,a2) -> real;
end;

:: XREAL_0:funcreg 34
registration
  let a1, a2 be real set;
  cluster max(a1,a2) -> real;
end;