let x, y, c be set ; :: thesis:
assume that
A1: x <> [<*y,c*>,'&'] and
A2: y <> [<*c,x*>,'&'] and
A3: c <> [<*x,y*>,'&'] ; :: thesis:
let s be State of (MajorityCirc (x,y,c)); :: thesis:
let a1, a2, a3 be Element of BOOLEAN ; :: thesis:
assume that
A4: a1 = s . x and
A5: a2 = s . y and
A6: a3 = s . c ; :: thesis:
set xy = [<*x,y*>,'&'];
set yc = [<*y,c*>,'&'];
set cx = [<*c,x*>,'&'];
set S = MajorityStr (x,y,c);
reconsider x9 = x, y9 = y, c9 = c as Vertex of (MajorityStr (x,y,c)) by FACIRC_1:72;
A7: InputVertices (MajorityStr (x,y,c)) = {x,y,c} by A1, A2, A3, Th20;
then A8: x in InputVertices (MajorityStr (x,y,c)) by ENUMSET1:def 1;
A9: y in InputVertices (MajorityStr (x,y,c)) by A7, ENUMSET1:def 1;
A10: c in InputVertices (MajorityStr (x,y,c)) by A7, ENUMSET1:def 1;
A11: (Following s) . x9 = s . x by A8, CIRCUIT2:def 5;
A12: (Following s) . y9 = s . y by A9, CIRCUIT2:def 5;
A13: (Following s) . c9 = s . c by A10, CIRCUIT2:def 5;
A14: Following (s,2) = Following (Following s) by FACIRC_1:15;
A15: (Following s) . [<*x,y*>,'&'] = a1 '&' a2 by A4, A5, A6, Lm3;
A16: (Following s) . [<*y,c*>,'&'] = a2 '&' a3 by A4, A5, A6, Lm3;
(Following s) . [<*c,x*>,'&'] = a3 '&' a1 by A4, A5, A6, Lm3;
hence (Following (s,2)) . (MajorityOutput (x,y,c)) = ((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1) by A14, A15, A16, Th28; :: thesis:
thus ( (Following (s,2)) . [<*x,y*>,'&'] = a1 '&' a2 & (Following (s,2)) . [<*y,c*>,'&'] = a2 '&' a3 & (Following (s,2)) . [<*c,x*>,'&'] = a3 '&' a1 ) by A4, A5, A6, A11, A12, A13, A14, Lm3; :: thesis: