per cases ( len R = 0 or len R <> 0 ) ;

suppose A1: len R = 0 ; :: thesis: ex b₁ being FinSequence of REAL st
( len b₁ = len R & b₁ . (len b₁) = R . (len R) & ( for n being Nat st 1 <= n & n <= (len b₁) - 1 holds
b₁ . n = (R . n) - (R . (n + 1)) ) )

take a = R; :: thesis: ( len a = len R & a . (len a) = R . (len R) & ( for n being Nat st 1 <= n & n <= (len a) - 1 holds
a . n = (R . n) - (R . (n + 1)) ) )
thus ( len a = len R & a . (len a) = R . (len R) ) ; :: thesis: for n being Nat st 1 <= n & n <= (len a) - 1 holds
a . n = (R . n) - (R . (n + 1))
let n be Nat; :: thesis: ( 1 <= n & n <= (len a) - 1 implies a . n = (R . n) - (R . (n + 1)) )
assume ( 1 <= n & n <= (len a) - 1 ) ; :: thesis: a . n = (R . n) - (R . (n + 1))
hence a . n = (R . n) - (R . (n + 1)) by A1, XXREAL_0:2; :: thesis: verum

end;

suppose len R <> 0 ; :: thesis: ex b₁ being FinSequence of REAL st
( len b₁ = len R & b₁ . (len b₁) = R . (len R) & ( for n being Nat st 1 <= n & n <= (len b₁) - 1 holds
b₁ . n = (R . n) - (R . (n + 1)) ) )

then 0 < len R by NAT_1:3;
then 0 + 1 <= len R by NAT_1:13;
then max (0,((len R) - 1)) = (len R) - 1 by FINSEQ_2:4;
then reconsider l = (len R) - 1 as Element of NAT by FINSEQ_2:5;
defpred S₂[ Nat, set ] means $2 = (R . $1) - (R . ($1 + 1));
set t = R . (len R);
A2: for n being Nat st n in Seg l holds
ex x being Real st S₂[n,x] ;
consider p being FinSequence of REAL such that
A3: ( dom p = Seg l & ( for n being Nat st n in Seg l holds
S₂[n,p . n] ) ) from FINSEQ_1:sch 5(A2);
take a = p ^ <*(R . (len R))*>; :: thesis: ( len a = len R & a . (len a) = R . (len R) & ( for n being Nat st 1 <= n & n <= (len a) - 1 holds
a . n = (R . n) - (R . (n + 1)) ) )
thus A4: len a = (len p) + (len <*(R . (len R))*>) by FINSEQ_1:22
.= l + (len <*(R . (len R))*>) by A3, FINSEQ_1:def 3
.= l + 1 by FINSEQ_1:39
.= len R ; :: thesis: ( a . (len a) = R . (len R) & ( for n being Nat st 1 <= n & n <= (len a) - 1 holds
a . n = (R . n) - (R . (n + 1)) ) )
hence a . (len a) = a . (l + 1)
.= a . ((len p) + 1) by A3, FINSEQ_1:def 3
.= R . (len R) by FINSEQ_1:42 ;
:: thesis: for n being Nat st 1 <= n & n <= (len a) - 1 holds
a . n = (R . n) - (R . (n + 1))
let n be Nat; :: thesis: ( 1 <= n & n <= (len a) - 1 implies a . n = (R . n) - (R . (n + 1)) )
assume ( 1 <= n & n <= (len a) - 1 ) ; :: thesis: a . n = (R . n) - (R . (n + 1))
then A5: n in Seg l by A4, FINSEQ_1:1;
then p . n = (R . n) - (R . (n + 1)) by A3;
hence a . n = (R . n) - (R . (n + 1)) by A3, A5, FINSEQ_1:def 7; :: thesis: verum

end;

end;