reserve A for non empty closed_interval Subset of REAL;

theorem Th21:
for a,b being Real holds (AffineMap (a,b)) is Lipschitzian
proof
 let a,b being Real;
 set f = AffineMap (a,b);
 ex r being Real st
 ( 0 < r &  ( for x1, x2 being Real st x1 in dom f & x2 in dom f holds
  |. (f . x1) - (f . x2) .| <= r * |. x1 - x2.| ) )
 proof
  take |. a .|+1;
 |. a .| >= 0 by COMPLEX1:46;
  hence |. a .|+1 > 0;
  thus for x1, x2 being Real st x1 in dom f & x2 in dom f holds
  |. (f . x1) - (f . x2) .| <= (|. a .|+1) * |. x1 - x2.|
  proof
   let x1, x2 be Real;
   assume x1 in dom f & x2 in dom f;
   B2: |. a .|+1 > |. a .|+0 by XREAL_1:8;
   BB: |.(x1-x2).| >= 0 by COMPLEX1:46;
   |. (f . x1) - (f . x2) .| =|.  a*x1+b- (f . x2) .| by FCONT_1:def 4
   .= |.  a*x1+b- (a*x2+b) .| by FCONT_1:def 4
   .= |.a*(x1-x2).|
   .= |.a.|*|.(x1-x2).| by COMPLEX1:65;
   hence thesis by XREAL_1:64,B2,BB;
  end;
 end;
 hence thesis;
end;
