
theorem Th39:
  CarProd(Seg 2 --> REAL) is Function of [:REAL,REAL:],REAL 2
& for s,t be object st s in REAL & t in REAL holds
          (CarProd(Seg 2 --> REAL)).([s,t])= <*s,t*>
proof
    set Y = Seg 2 --> REAL;
    set Y1 = SubFin(Y,1);
    set F = CarProd(Seg 2 --> REAL);

    thus F is Function of [:REAL,REAL:],REAL 2 by Th37,SRINGS_5:8;
    thus for s,t be object st s in REAL & t in REAL holds F.([s,t]) = <*s,t*>
    proof
     let s,t be object;
     assume
A1:  s in REAL & t in REAL;

A2:  1 in Seg 1 & 1 in Seg 2 & 2 in Seg 2;

A3:  Y1 = Y|1 by MEASUR13:def 5; then
A4:  Y1.1 = Y.1 by A2,FUNCT_1:49;
     CarProduct Y1 = (Y|1).1 by A3,MEASUR13:def 3; then
A5:  CarProduct Y1 = REAL by A3,A4,A2,FUNCOP_1:7;

     Y1 = <*REAL*> by A2,A3,A4,FUNCOP_1:7,FINSEQ_1:40; then
A6:  Y1 = 1 |-> REAL by FINSEQ_2:59;

     ElmFin(Y,1+1) = REAL by Th37; then
     ex u,v be FinSequence st
      (CarProd Y1).s = u & <*t*> = v & (CarProd Y).(s,t) = u^v by A5,A1,Th13;
     hence (CarProd Y).([s,t]) = <*s,t*> by A1,A6,Th38;
    end;
end;
