reserve p for Real;
reserve S,T for RealNormSpace;
reserve x0 for Point of S;
reserve f for PartFunc of S,T;
reserve c for constant sequence of S;
reserve R for RestFunc of S,T;

theorem Th4:
  (for X be RealNormSpace, x,y be Point of X holds ||.x-y.|| > 0
iff x <> y) &
(for X be RealNormSpace, x,y be Point of X holds ||.x-y.|| = ||.y
-x.||) &
(for X be RealNormSpace, x,y be Point of X holds ||.x-y.|| = 0 iff x =
y) & (for X be RealNormSpace, x,y be Point of X holds ||.x-y.|| <> 0 iff x <> y
) & (for X be RealNormSpace, x,y,z be Point of X,
  e be Real st e > 0 holds ( (
  ||.x-z.|| <e/2 & ||.z-y.|| <e/2) implies ||.x-y.|| <e) ) & (for X be
RealNormSpace, x,y,z be Point of X, e be Real st
e > 0 holds ((||.x-z.|| <e/2
& ||.y-z.|| <e/2) implies ||.x-y.|| <e) ) & (for X be RealNormSpace, x be Point
  of X st ( for e be Real st e>0
  holds ||.x.|| <e ) holds x=0.X ) & for X be
RealNormSpace, x,y be Point of X st ( for e be Real
st e>0 holds ||.x-y.|| <e )
  holds x=y
proof
  thus for X be RealNormSpace, x,y be Point of X holds ||.x-y.|| > 0 iff x <> y
  proof
    let X be RealNormSpace;
    let x, y be Point of X;
    0 < ||.x-y.|| implies x-y <> 0.X by NORMSP_0:def 6;
    hence 0 < ||.x-y.|| implies x <> y by RLVECT_1:15;
    now
      assume x <> y;
      then 0 <> ||.x-y.|| by NORMSP_1:11;
      hence 0 < ||.x-y.|| by NORMSP_1:4;
    end;
    hence thesis;
  end;
  thus for X be RealNormSpace, x,y be Point of X holds ||.x-y.|| = ||.y-x.||
  proof
    let X be RealNormSpace;
    let x, y be Point of X;
    thus ||.x-y.|| = ||.-(x-y).|| by NORMSP_1:2
      .= ||.y-x.|| by RLVECT_1:33;
  end;
A1: for X be RealNormSpace for x, y, z be Point of X
  for e be Real st e > 0
  holds ( ||.x-z.|| < e/2 & ||.z-y.|| < e/2 implies ||.x-y.|| <e)
  proof
    let X be RealNormSpace;
    let x, y, z be Point of X;
    let e be Real such that
    e > 0;
    assume ||.x-z.|| < e/2 & ||.z-y.|| < e/2;
    then ||.x-z.|| + ||.z-y.|| < e/2 + e/2 by XREAL_1:8;
    then ||.x-y.|| + (||.x-z.|| + ||.z-y.||) < (||.x-z.|| + ||.z-y.||) + e by
NORMSP_1:10,XREAL_1:8;
    then
    ||.x-y.|| + (||.x-z.|| + ||.z-y.||) +- (||.x-z.|| + ||.z-y.||) < e + (
    ||.x-z.|| + ||.z-y.||) +- (||.x-z.|| + ||.z-y.||) by XREAL_1:8;
    hence thesis;
  end;
A2: for X be RealNormSpace for x, y, z be Point of X
  for e be Real st e > 0
  holds ( ||.x-z.|| < e/2 & ||.y-z.|| < e/2 implies ||.x-y.|| < e)
  proof
    let X be RealNormSpace;
    let x, y, z be Point of X;
    let e be Real such that
A3: e > 0;
    assume that
A4: ||.x-z.|| < e/2 and
A5: ||.y-z.|| < e/2;
    ||.-(y-z).|| < e/2 by A5,NORMSP_1:2;
    then ||.z-y.|| < e/2 by RLVECT_1:33;
    hence thesis by A1,A3,A4;
  end;
A6: for X be RealNormSpace for x be Point of X st
   (for e be Real st e>0
  holds ||.x.|| <e) holds x=0.X
  proof
    let X be RealNormSpace;
    let x be Point of X such that
A7: for e be Real st e > 0 holds ||.x.|| <e;
    now
      assume x<>0.X;
      then 0 <> ||.x.|| by NORMSP_0:def 5;
      then 0 < ||.x.|| by NORMSP_1:4;
      hence contradiction by A7;
    end;
    hence thesis;
  end;
  for X be RealNormSpace for x,y be Point of X st
  (for e be Real st e>0 holds ||.x-y.|| <e) holds x=y
  proof
    let X be RealNormSpace;
    let x,y be Point of X;
    assume for e be Real st e > 0 holds ||.x-y.|| <e;
    then x-y = 0.X by A6;
    hence thesis by RLVECT_1:21;
  end;
  hence thesis by A1,A2,A6,NORMSP_1:6;
end;
