reserve n   for Nat,
        r,s for Real,
        x,y for Element of REAL n,
        p,q for Point of TOP-REAL n,
        e   for Point of Euclid n;
reserve n for non zero Nat;
reserve n for non zero Nat;
reserve n for Nat,
        X for set,
        S for Subset-Family of X;
reserve n for Nat,
        S for Subset-Family of REAL;
reserve n       for Nat,
        a,b,c,d for Element of REAL n;
reserve n for non zero Nat;
reserve n     for non zero Nat,
        x,y,z for Element of REAL n;

theorem Th61:
  (Infty_dist n).(x,y) <= (Infty_dist n).(x,z) + (Infty_dist n).(z,y)
  proof
    (Infty_dist n).(x,y) in REAL & (Infty_dist n).(x,z) in REAL &
    (Infty_dist n).(z,y) in REAL;
    then reconsider sxy = sup rng abs(x-y), sxz = sup rng abs(x-z),
    szy = sup rng abs(z-y) as Real by Def7;
    sxy <= sxz + szy
    proof
      for er be ExtReal st er in rng abs(x-y) holds er <= sxz + szy
      proof
        let er be ExtReal;
        assume er in rng abs(x-y);
        then consider i be object such that
A1:     i in dom abs(x-y) and
A2:     er = abs(x-y).i by FUNCT_1:def 3;
        abs(x-y).i <= sxz + szy
        proof
A3:       abs(x-y).i <= abs(x-z).i + abs(z-y).i
          proof
            reconsider fxy = x-y,fxz = x - z,
                       fzy = z - y as complex-valued Function;
A4:         abs fxy.i = |.(x-y).i.| & abs fxz.i = |.(x-z).i.| &
            abs fzy.i = |.(z-y).i.| by VALUED_1:18;
            |.(x-y).i.| <= |.(x-z).i.| + |.(z-y).i.|
            proof
A5:           |.(x-y).i.| = |.x.i - y.i.| & |.(x-z).i.| = |.x.i-z.i.| &
                |.(z-y).i.| = |.z.i-y.i.| by A1,RVSUM_1:27;
              |.(x.i - z.i) + (z.i - y.i).| <= |.x.i-z.i.| + |.z.i-y.i.|
                by COMPLEX1:56;
              hence thesis by A5;
            end;
            hence thesis by A4;
          end;
          abs(x-z).i + abs(z-y).i <= sxz + szy
          proof
A6:         dom abs(x-z) = Seg n & dom abs (z-y) = Seg n & dom abs(x-y) = Seg n
              by FINSEQ_2:124;
            abs(x-z).i in rng abs(x-z) & abs(z-y).i in rng abs(z-y)
              by A6,A1,FUNCT_1:def 3; then
A7:         abs(x-z).i <= sup rng abs(x-z) & abs(z-y).i <= sup rng abs (z-y)
              by XXREAL_2:4;
            abs(x-z).i in REAL & abs(z-y).i in REAL by A1,A6,Th2; then
            abs(x-z).i + abs(z-y).i <= sup rng abs(x-z) + sup rng abs (z-y)
             by Th3,A7;
            hence thesis by XXREAL_3:def 2;
          end;
          hence thesis by A3,XXREAL_0:2;
        end;
        hence thesis by A2;
      end;
      then sxz + szy is UpperBound of rng abs(x-y) by XXREAL_2:def 1;
      hence thesis by XXREAL_2:def 3;
    end;
    then sxy <= sxz + (Infty_dist n).(z,y) by Def7;
    then sxy <= (Infty_dist n).(x,z) + (Infty_dist n).(z,y) by Def7;
    hence thesis by Def7;
  end;
