
theorem Th38:
  CarProd(Seg 1 --> REAL) is Function of REAL,REAL 1
& for s be object st s in REAL holds (CarProd(Seg 1 --> REAL)).s = <*s*>
proof
    set X = Seg 1 --> REAL;
    set E = CarProd(Seg 1 --> REAL);
    thus E is Function of REAL,REAL 1 by Th37,SRINGS_5:8;

    thus for s be object st s in REAL holds E.s = <*s*>
    proof
     let s be object;
     assume
A1:  s in REAL;
A2:  ex id1 be Function of CarProduct SubFin(X,1),product SubFin(X,1) st
      (Pt2FinSeq X).1 = id1 & id1 is bijective
    & for x be object st x in CarProduct SubFin(X,1) holds id1.x = <*x*>
        by Def5;
     SubFin(X,1) = X|1 by MEASUR13:def 5;
     hence E.s = <*s*> by A1,A2,Th37;
    end;
end;
