theorem (U-concatenation).:(id (1-tuples_on U)) =
the set of all <*u,u*> where u is Element of U ::#Th38
proof
set f=U-concatenation, U1=1-tuples_on U, D=id U1, U2=2-tuples_on U, A=f.:D,
B= the set of all <*u,u*> where u is Element of U;
D c= [:U1, U1:] & U1 c= U* by FINSEQ_2:142; then
[:U1, U1:] c= [:U*, U*:] by ZFMISC_1:96; then
reconsider DD=D as Subset of [:U*, U*:] by XBOOLE_1:1;
A1: U1=the set of all <*u*> where u is Element of U by FINSEQ_2:96;
A2: dom D=U1 & dom f=[:U*, U*:] by FUNCT_2:def 1; then
A3: D={[x,D.x] where x is Element of U1: x in U1} by Th20;
now
let y be object; assume y in A; then consider x being object such that
A4: x in dom f & x in D & y=f.x by FUNCT_1:def 6;
consider p being Element of U1 such that
A5: x=[p,D.p] & p in U1 by A4, A3; consider u being Element of U such that
A6: p=<*u*> by A5, A1;
reconsider pp=p as FinSequence of U;
y=f.(pp,pp) by A4, A5; then
y = <*u,u*> by Th4, A6; hence y in B;
end; then
A7: A c= B;
now
let y be object; assume y in B; then consider u being Element of U such that
A8: y=<*u,u*>; reconsider p=<*u*> as Element of
U1 by FINSEQ_2:98; reconsider pp=p as FinSequence of U;
[p,D.p]=[p,p] & p in U1; then [p,p] in D by A3; then
reconsider dd=[p,p] as Element of D;
A9: dd in DD null [:U*, U*:];  y = f.(pp,pp) by A8, Th4 .= f.dd;
hence y in f.:D by A9, A2, FUNCT_1:def 6;
end;
then B c= A; hence thesis by A7;
end;
