Article HEINE, MML version 4.99.1005

:: HEINE:th 1
theorem
for b1, b2 being real set
for b3 being SubSpace of RealSpace
for b4, b5 being Element of the carrier of b3
      st b1 = b4 & b2 = b5
   holds dist(b4,b5) = abs (b1 - b2);

:: HEINE:th 2
theorem
for b1, b2, b3 being real set
      st b1 <= b2 & b2 <= b3
   holds [.b1,b2.] \/ [.b2,b3.] = [.b1,b3.];

:: HEINE:th 6
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is increasing & proj2 b2 c= NAT
   holds b1 <= b2 . b1;

:: HEINE:funcnot 1 => HEINE:func 1
definition
  let a1 be Function-like quasi_total Relation of NAT,REAL;
  let a2 be Element of NAT;
  func A2 to_power A1 -> Function-like quasi_total Relation of NAT,REAL means
    for b1 being Element of NAT holds
       it . b1 = a2 to_power (a1 . b1);
end;

:: HEINE:def 1
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of NAT
for b3 being Function-like quasi_total Relation of NAT,REAL holds
      b3 = b2 to_power b1
   iff
      for b4 being Element of NAT holds
         b3 . b4 = b2 to_power (b1 . b4);

:: HEINE:th 9
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is divergent_to+infty
   holds 2 to_power b1 is divergent_to+infty;

:: HEINE:th 11
theorem
for b1, b2 being real set
      st b1 <= b2
   holds Closed-Interval-TSpace(b1,b2) is compact;