
theorem Th37:
  CarProduct(Seg 1 --> REAL) = REAL
& ElmFin(Seg 1 --> REAL,1) = REAL
& CarProduct(Seg 2 --> REAL) = [:REAL,REAL:]
& ElmFin(Seg 2 --> REAL,2) = REAL
& CarProduct(Seg 3 --> REAL) = [:REAL,REAL,REAL:]
proof
    set X = Seg 1 --> REAL;
    1 in Seg 1; then
    X.1 = REAL by FUNCOP_1:7;
    hence CarProduct X = REAL & ElmFin(Seg 1 --> REAL,1) = REAL
      by MEASUR13:def 3,def 1;

    set Y = Seg 2 --> REAL;
A1: 1 in Seg 2 & 2 in Seg 2; then
A2: Y.1 = REAL & Y.2 = REAL by FUNCOP_1:7;

    CarProduct Y = [: (ProdFinSeq Y).1,Y.(1+1) :] by MEASUR13:def 3;
    hence CarProduct Y = [: REAL,REAL :] by A2,MEASUR13:def 3;
    ElmFin(Y,2) = Y.2 by MEASUR13:def 1;
    hence ElmFin(Y,2) = REAL by A1,FUNCOP_1:7;

    set Z = Seg 3 --> REAL;
    1 in Seg 3 & 2 in Seg 3 & 3 in Seg 3; then
A3: Z.1 = REAL & Z.2 = REAL & Z.3 = REAL by FUNCOP_1:7;
    (ProdFinSeq Z).1 = Z.1
  & (ProdFinSeq Z).2 = [: (ProdFinSeq Z).1,Z.(1+1) :]
  & (ProdFinSeq Z).3 = [: (ProdFinSeq Z).2,Z.(2+1) :] by MEASUR13:def 3;
    hence CarProduct Z = [: REAL,REAL,REAL :] by A3,ZFMISC_1:def 3;
end;
