let z1 be Tuple of 2, BOOLEAN ; :: thesis: ( z1 = <*FALSE *> ^ <*TRUE *> implies Intval z1 = - 2 )
assume A1: z1 = <*FALSE *> ^ <*TRUE *> ; :: thesis: Intval z1 = - 2
consider k1, k2 being Element of NAT such that
A2: Binary z1 = <*k1,k2*> by FINSEQ_2:120;
A3: z1 = <*FALSE ,TRUE *> by A1, FINSEQ_1:def 9;
A4: z1 /. 1 = FALSE by A3, FINSEQ_4:26;
A5: z1 /. 2 = TRUE by A3, FINSEQ_4:26;
A6: Intval z1 = (Absval z1) - (2 to_power (1 + 1)) by A5, Def3
.= (Absval z1) - ((2 to_power 1) * (2 to_power 1)) by POWER:32
.= (Absval z1) - (2 * (2 to_power 1)) by POWER:30
.= (Absval z1) - (2 * 2) by POWER:30
.= (Absval z1) - 4 ;
A7: ( 1 in Seg 1 & Seg 1 c= Seg 2 ) by FINSEQ_1:5, FINSEQ_1:7;
A8: (Binary z1) /. 1 = IFEQ (z1 /. 1),FALSE ,0 ,(2 to_power (1 -' 1)) by A7, BINARITH:def 6
.= 0 by A4, FUNCOP_1:def 8 ;
A9: 2 in Seg 2 by FINSEQ_1:5;
A10: (Binary z1) /. 2 = IFEQ (z1 /. 2),FALSE ,0 ,(2 to_power (2 -' 1)) by A9, BINARITH:def 6
.= 2 to_power (2 -' 1) by A5, FUNCOP_1:def 8 ;
A11: 2 - 1 = 1 ;
A12: 2 -' 1 = 1 by A11, XREAL_0:def 2;
A13: (Binary z1) /. 2 = 2 by A10, A12, POWER:30;
A14: ( (Binary z1) /. 1 = k1 & (Binary z1) /. 2 = k2 ) by A2, FINSEQ_4:26;
A15: Absval z1 = addnat $$ <*0 ,2*> by A2, A8, A13, A14, BINARITH:def 7
.= addnat $$ (<*0 *> ^ <*2*>) by FINSEQ_1:def 9
.= addnat . (addnat $$ <*0 *>),(addnat $$ <*2*>) by FINSOP_1:6
.= addnat . 0 ,(addnat $$ <*2*>) by FINSOP_1:12
.= addnat . 0 ,2 by FINSOP_1:12
.= 0 + 2 by BINOP_2:def 23
.= 2 ;
thus Intval z1 = - 2 by A6, A15; :: thesis: verum