let z1 be Tuple of 2,BOOLEAN ; :: thesis: ( z1 = <*TRUE *> ^ <*FALSE *> implies Intval z1 = 1 )
assume A1: z1 = <*TRUE *> ^ <*FALSE *> ; :: thesis: Intval z1 = 1
consider k1, k2 being Element of NAT such that
A2: Binary z1 = <*k1,k2*> by FINSEQ_2:120;
z1 = <*TRUE ,FALSE *> by A1, FINSEQ_1:def 9;
then A3: ( z1 /. 1 = TRUE & z1 /. 2 = FALSE ) by FINSEQ_4:26;
A4: 1 in Seg 1 by FINSEQ_1:5;
Seg 1 c= Seg 2 by FINSEQ_1:7;
then A5: (Binary z1) /. 1 = IFEQ (z1 /. 1),FALSE ,0 ,(2 to_power (1 -' 1)) by A4, BINARITH:def 6
.= 2 to_power (1 -' 1) by A3, FUNCOP_1:def 8 ;
1 - 1 = 0 ;
then 1 -' 1 = 0 by XREAL_0:def 2;
then A6: (Binary z1) /. 1 = 1 by A5, POWER:29;
A7: (Binary z1) /. 1 = k1 by A2, FINSEQ_4:26;
2 in Seg 2 by FINSEQ_1:5;
then A8: (Binary z1) /. 2 = IFEQ (z1 /. 2),FALSE ,0 ,(2 to_power (2 -' 1)) by BINARITH:def 6
.= 0 by A3, FUNCOP_1:def 8 ;
(Binary z1) /. 2 = k2 by A2, FINSEQ_4:26;
then Absval z1 = addnat $$ <*1,0 *> by A2, A6, A7, A8, BINARITH:def 7
.= addnat $$ (<*1*> ^ <*0 *>) by FINSEQ_1:def 9
.= addnat . (addnat $$ <*1*>),(addnat $$ <*0 *>) by FINSOP_1:6
.= addnat . 1,(addnat $$ <*0 *>) by FINSOP_1:12
.= addnat . 1,0 by FINSOP_1:12
.= 1 + 0 by BINOP_2:def 23
.= 1 ;
hence Intval z1 = 1 by A3, Def3; :: thesis: verum