reserve A for non empty closed_interval Subset of REAL;

theorem Th17X:
for a,b,p,q being Real, f be Function of REAL,REAL st a <> p &
f = ( AffineMap (a,b) | ].-infty,(q-b)/(a-p).[ )
 +* ( AffineMap (p,q) | [.(q-b)/(a-p),+infty.[ )
holds
f is_integrable_on A & f | A is bounded
proof
 let a,b,p,q being Real, f be Function of REAL,REAL;
 assume that
 A1:  a <> p and
 A2: f = ( AffineMap (a,b) | ].-infty,(q-b)/(a-p).[ )
      +* ( AffineMap (p,q) | [.(q-b)/(a-p),+infty.[ );
 reconsider f as PartFunc of REAL,REAL;
 A3: REAL = dom f by FUNCT_2:def 1;
 f  is Lipschitzian by Th17L,A1,A2;
 then
 f | A is continuous;
 hence thesis by INTEGRA5:10,INTEGRA5:11,A3;
end;
