Article ALI2, MML version 4.99.1005

:: ALI2:modenot 1 => ALI2:mode 1
definition
  let a1 be non empty Reflexive discerning symmetric triangle MetrStruct;
  mode contraction of A1 -> Function-like quasi_total Relation of the carrier of a1,the carrier of a1 means
    ex b1 being Element of REAL st
       0 < b1 &
        b1 < 1 &
        (for b2, b3 being Element of the carrier of a1 holds
        dist(it . b2,it . b3) <= b1 * dist(b2,b3));
end;

:: ALI2:dfs 1
definiens
  let a1 be non empty Reflexive discerning symmetric triangle MetrStruct;
  let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of a1;
To prove
     a2 is contraction of a1
it is sufficient to prove
  thus ex b1 being Element of REAL st
       0 < b1 &
        b1 < 1 &
        (for b2, b3 being Element of the carrier of a1 holds
        dist(a2 . b2,a2 . b3) <= b1 * dist(b2,b3));

:: ALI2:def 1
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 holds
      b2 is contraction of b1
   iff
      ex b3 being Element of REAL st
         0 < b3 &
          b3 < 1 &
          (for b4, b5 being Element of the carrier of b1 holds
          dist(b2 . b4,b2 . b5) <= b3 * dist(b4,b5));

:: ALI2:th 2
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being contraction of b1
      st TopSpaceMetr b1 is compact
   holds ex b3 being Element of the carrier of b1 st
      b2 . b3 = b3 &
       (for b4 being Element of the carrier of b1
             st b2 . b4 = b4
          holds b4 = b3);