reserve a,b,r for Real;
reserve A for non empty set;
reserve X,x for set;
reserve f,g,F,G for PartFunc of REAL,REAL;
reserve n for Element of NAT;

theorem Th27:
  exp_R is_integral_of exp_R,REAL
proof
A1: dom(exp_R|REAL) = REAL /\ REAL by TAYLOR_1:16;
A2: now
    let x be object;
    assume
A3: x in dom(exp_R`|REAL);
    then reconsider z=x as Real;
    reconsider z1=z as Element of REAL by XREAL_0:def 1;
    (exp_R`|REAL).x = diff(exp_R,z) by A3,FDIFF_1:def 7,TAYLOR_1:16;
    then (exp_R`|REAL).x = exp_R.z1 by TAYLOR_1:16;
    hence (exp_R`|REAL).x = (exp_R|REAL).x;
  end;
  dom(exp_R`|REAL) = REAL by FDIFF_1:def 7,TAYLOR_1:16;
  then exp_R`|REAL=exp_R|REAL by A1,A2,FUNCT_1:2;
  hence thesis by Lm1,TAYLOR_1:16;
end;
