reserve A for non empty closed_interval Subset of REAL;

theorem
for c being Real, f,g be Function of REAL,REAL st
f is continuous & g is continuous & f . c = g . c holds
(f | ].-infty,c.]) +* (g | [.c,+infty.[) is continuous Function of REAL,REAL
proof
 let c be Real, f,g be Function of REAL,REAL;
 assume that
 A1: f is continuous & g is continuous and
 A2: f . c = g . c;
 reconsider F = (f|].-infty,c.]) +* (g|[.c,+infty.[) as PartFunc of REAL,REAL
  by FUZZY_7:12;
 ].-infty,c.] c= REAL & [.c,+infty.[ c= REAL; then
 ].-infty,c.] c= dom f & [.c,+infty.[ c= dom g by FUNCT_2:def 1;
 hence thesis by Th1,A2,A1,FUZZY_7:12;
end;
