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);