
theorem
  CarProd(Seg 3 --> REAL) is Function of [:REAL,REAL,REAL:],REAL 3
& for s,t,u be object st s in REAL & t in REAL & u in REAL holds
          (CarProd(Seg 3 --> REAL)). [[s,t],u] = <*s,t,u*>
proof
    set Z = Seg 3 --> REAL;
    set Z2 = SubFin(Z,2);
    set H = CarProd(Seg 3 --> REAL);
    thus H is Function of [:REAL,REAL,REAL:],REAL 3 by Th37,SRINGS_5:8;
    thus for s,t,u be object st s in REAL & t in REAL & u in REAL holds
          H.([[s,t],u]) = <*s,t,u*>
    proof
     let s,t,u be object;
     assume
A1:  s in REAL & t in REAL & u in REAL;

A2:  1 in Seg 2 & 2 in Seg 2 & 1 in Seg 3 & 2 in Seg 3 & 3 in Seg 3;

A3:  len Z2 = 2 by CARD_1:def 7;
A4:  Z2 = Z|2 by MEASUR13:def 5; then
     Z2.2 = Z.2 by A2,FUNCT_1:49; then
A5:  Z2.2 = REAL by A2,FUNCOP_1:7;

     Z2.1 = Z.1 by A4,A2,FUNCT_1:49; then
     Z2 = <*REAL,REAL*> by A5,A3,A2,FUNCOP_1:7,FINSEQ_1:44; then
A6:  Z2 = 2 |-> REAL by FINSEQ_2:61; then
A7:  (CarProd Z2).([s,t]) = <*s,t*> by A1,Th39;

     ElmFin(Z,3) = Z.3 by MEASUR13:def 1; then
     ElmFin(Z,2+1) = REAL by A2,FUNCOP_1:7; then
     ex a,b be FinSequence st
     (CarProd Z2).([s,t]) = a & <*u*> = b  & (CarProd Z).([s,t],u) = a^b
        by A1,A6,Th37,Th13,ZFMISC_1:87;
     hence (CarProd Z).([[s,t],u]) = <*s,t,u*> by A7;
    end;
end;
