let a1, a2 be Element of SCM+FSA-Data*-Loc ; :: thesis: ( ex c being Element of SCM-Data-Loc ex f being Element of SCM+FSA-Data*-Loc st

( <*c,f*> = x `3_3 & a1 = f ) & ex c being Element of SCM-Data-Loc ex f being Element of SCM+FSA-Data*-Loc st

( <*c,f*> = x `3_3 & a2 = f ) implies a1 = a2 )

given c1 being Element of SCM-Data-Loc , f1 being Element of SCM+FSA-Data*-Loc such that A5: <*c1,f1*> = x `3_3 and

A6: a1 = f1 ; :: thesis: ( for c being Element of SCM-Data-Loc

for f being Element of SCM+FSA-Data*-Loc holds

( not <*c,f*> = x `3_3 or not a2 = f ) or a1 = a2 )

given c2 being Element of SCM-Data-Loc , f2 being Element of SCM+FSA-Data*-Loc such that A7: ( <*c2,f2*> = x `3_3 & a2 = f2 ) ; :: thesis: a1 = a2

thus a1 = <*c1,f1*> . 2 by A6, FINSEQ_1:44

.= a2 by A5, A7, FINSEQ_1:44 ; :: thesis: verum

( <*c,f*> = x `3_3 & a1 = f ) & ex c being Element of SCM-Data-Loc ex f being Element of SCM+FSA-Data*-Loc st

( <*c,f*> = x `3_3 & a2 = f ) implies a1 = a2 )

given c1 being Element of SCM-Data-Loc , f1 being Element of SCM+FSA-Data*-Loc such that A5: <*c1,f1*> = x `3_3 and

A6: a1 = f1 ; :: thesis: ( for c being Element of SCM-Data-Loc

for f being Element of SCM+FSA-Data*-Loc holds

( not <*c,f*> = x `3_3 or not a2 = f ) or a1 = a2 )

given c2 being Element of SCM-Data-Loc , f2 being Element of SCM+FSA-Data*-Loc such that A7: ( <*c2,f2*> = x `3_3 & a2 = f2 ) ; :: thesis: a1 = a2

thus a1 = <*c1,f1*> . 2 by A6, FINSEQ_1:44

.= a2 by A5, A7, FINSEQ_1:44 ; :: thesis: verum