reserve A for non empty closed_interval Subset of REAL;

theorem Th17L:
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 Lipschitzian
proof
 let a,b,p,q being Real, f be Function of REAL,REAL;
 assume that
 AP: a <> p and
 AF: f = ( AffineMap (a,b) | ].-infty,(q-b)/(a-p).[ )
      +* ( AffineMap (p,q) | [.(q-b)/(a-p),+infty.[ );
 set fa = AffineMap (a,b);
 set fp = AffineMap (p,q);
 B1: a-p <> 0 by AP;
 A1: fa is Lipschitzian by Th21;
 A2: fp is Lipschitzian by Th21;
 B2: fa.((q-b)/(a-p)) = a*((q-b)/(a-p))+b by FCONT_1:def 4
 .= a*(q-b)/(a-p)+b by XCMPLX_1:74
 .= (a*(q-b)+(a-p)*b)/(a-p) by XCMPLX_1:113,B1;
 fp.((q-b)/(a-p)) = p*((q-b)/(a-p))+q by FCONT_1:def 4
 .= p*(q-b)/(a-p)+q by XCMPLX_1:74
 .= (p*(q-b)+(a-p)*q)/(a-p) by XCMPLX_1:113,B1;
 hence thesis by Th20,AF,A1,A2,B2;
end;
