reserve r,p,x for Real;
reserve n for Element of NAT;
reserve A for non empty closed_interval Subset of REAL;
reserve Z for open Subset of REAL;
reserve a,b,x for Real;
reserve n for Element of NAT;
reserve A for non empty closed_interval Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;
reserve Z for open Subset of REAL;

theorem
  not -a in A implies (AffineMap (1,a))^|A is continuous
proof
  set g2 = AffineMap (1,a);
  assume
A1: not -a in A;
  not 0 in rng (g2|A)
  proof
    set h2 = g2|A;
    assume 0 in rng (g2|A);
    then consider x being object such that
A2: x in dom h2 and
A3: h2.x = 0 by FUNCT_1:def 3;
    reconsider d = x as Element of REAL by A2;
A4: g2.d = a+1*d by FCONT_1:def 4;
    d in A by A2,RELAT_1:57;
    then dom h2 c= A & a+d = 0 by A3,A4,FUNCT_1:49,RELAT_1:58;
    hence thesis by A1,A2;
  end;
  then
A5: (g2|A)"{0} = {} by FUNCT_1:72;
  thus thesis by A5,FCONT_1:23;
end;
