let f be object ; :: according to FINSEQ_1:def 19 :: thesis: ( not f in proj2 SCMPDS-Instr or f is set )
assume f in proj2 SCMPDS-Instr ; :: thesis: f is set
then consider y being object such that
A1: [y,f] in SCMPDS-Instr by XTUPLE_0:def 13;
set x = [y,f];

per cases ( [y,f] in ((({[0,{},{}]} \/ { [14,{},<*l*>] where l is Element of INT : verum } ) \/ { [1,{},<*sp*>] where sp is Element of SCM-Data-Loc : verum } ) \/ { [I,{},<*v,c*>] where I is Element of Segm 15, v is Element of SCM-Data-Loc , c is Element of INT : I in {2,3} } ) \/ { [I,{},<*v,c1,c2*>] where I is Element of Segm 15, v is Element of SCM-Data-Loc , c1, c2 is Element of INT : I in {4,5,6,7,8} } or [y,f] in { [I,{},<*v1,v2,c1,c2*>] where I is Element of Segm 15, v1, v2 is Element of SCM-Data-Loc , c1, c2 is Element of INT : I in {9,10,11,12,13} } ) by A1, XBOOLE_0:def 3;

suppose A2: [y,f] in ((({[0,{},{}]} \/ { [14,{},<*l*>] where l is Element of INT : verum } ) \/ { [1,{},<*sp*>] where sp is Element of SCM-Data-Loc : verum } ) \/ { [I,{},<*v,c*>] where I is Element of Segm 15, v is Element of SCM-Data-Loc , c is Element of INT : I in {2,3} } ) \/ { [I,{},<*v,c1,c2*>] where I is Element of Segm 15, v is Element of SCM-Data-Loc , c1, c2 is Element of INT : I in {4,5,6,7,8} } ; :: thesis: f is set

per cases ( [y,f] in (({[0,{},{}]} \/ { [14,{},<*l*>] where l is Element of INT : verum } ) \/ { [1,{},<*sp*>] where sp is Element of SCM-Data-Loc : verum } ) \/ { [I,{},<*v,c*>] where I is Element of Segm 15, v is Element of SCM-Data-Loc , c is Element of INT : I in {2,3} } or [y,f] in { [I,{},<*v,c1,c2*>] where I is Element of Segm 15, v is Element of SCM-Data-Loc , c1, c2 is Element of INT : I in {4,5,6,7,8} } ) by A2, XBOOLE_0:def 3;

suppose [y,f] in (({[0,{},{}]} \/ { [14,{},<*l*>] where l is Element of INT : verum } ) \/ { [1,{},<*sp*>] where sp is Element of SCM-Data-Loc : verum } ) \/ { [I,{},<*v,c*>] where I is Element of Segm 15, v is Element of SCM-Data-Loc , c is Element of INT : I in {2,3} } ; :: thesis: f is set

then [y,f] in ({[0,{},{}]} \/ ( { [14,{},<*l*>] where l is Element of INT : verum } \/ { [1,{},<*sp*>] where sp is Element of SCM-Data-Loc : verum } )) \/ { [I,{},<*v,c*>] where I is Element of Segm 15, v is Element of SCM-Data-Loc , c is Element of INT : I in {2,3} } by XBOOLE_1:4;
then ( [y,f] in {[0,{},{}]} \/ ( { [14,{},<*l*>] where l is Element of INT : verum } \/ { [1,{},<*sp*>] where sp is Element of SCM-Data-Loc : verum } ) or [y,f] in { [I,{},<*v,c*>] where I is Element of Segm 15, v is Element of SCM-Data-Loc , c is Element of INT : I in {2,3} } ) by XBOOLE_0:def 3;

per cases then ( [y,f] in {[0,{},{}]} or [y,f] in { [14,{},<*l*>] where l is Element of INT : verum } \/ { [1,{},<*sp*>] where sp is Element of SCM-Data-Loc : verum } or [y,f] in { [I,{},<*v,c*>] where I is Element of Segm 15, v is Element of SCM-Data-Loc , c is Element of INT : I in {2,3} } ) by XBOOLE_0:def 3;

suppose [y,f] in {[0,{},{}]} ; :: thesis: f is set

then [y,f] = [0,{},{}] by TARSKI:def 1;
hence f is FinSequence by XTUPLE_0:1; :: thesis: verum

end;

suppose A3: [y,f] in { [14,{},<*l*>] where l is Element of INT : verum } \/ { [1,{},<*sp*>] where sp is Element of SCM-Data-Loc : verum } ; :: thesis: f is set

per cases ( [y,f] in { [14,{},<*l*>] where l is Element of INT : verum } or [y,f] in { [1,{},<*sp*>] where sp is Element of SCM-Data-Loc : verum } ) by A3, XBOOLE_0:def 3;

suppose [y,f] in { [14,{},<*l*>] where l is Element of INT : verum } ; :: thesis: f is set

then ex l being Element of INT st [y,f] = [14,{},<*l*>] ;
hence f is FinSequence by XTUPLE_0:1; :: thesis: verum

end;

suppose [y,f] in { [1,{},<*sp*>] where sp is Element of SCM-Data-Loc : verum } ; :: thesis: f is set

then ex sp being Element of SCM-Data-Loc st [y,f] = [1,{},<*sp*>] ;
hence f is FinSequence by XTUPLE_0:1; :: thesis: verum

end;

end;

end;

suppose [y,f] in { [I,{},<*v,c*>] where I is Element of Segm 15, v is Element of SCM-Data-Loc , c is Element of INT : I in {2,3} } ; :: thesis: f is set

then ex I being Element of Segm 15 ex v being Element of SCM-Data-Loc ex c being Element of INT st
( [y,f] = [I,{},<*v,c*>] & I in {2,3} ) ;
hence f is FinSequence by XTUPLE_0:1; :: thesis: verum

end;

end;

end;

suppose [y,f] in { [I,{},<*v,c1,c2*>] where I is Element of Segm 15, v is Element of SCM-Data-Loc , c1, c2 is Element of INT : I in {4,5,6,7,8} } ; :: thesis: f is set

then ex I being Element of Segm 15 ex v being Element of SCM-Data-Loc ex c1, c2 being Element of INT st
( [y,f] = [I,{},<*v,c1,c2*>] & I in {4,5,6,7,8} ) ;
hence f is FinSequence by XTUPLE_0:1; :: thesis: verum

end;

end;

end;

suppose [y,f] in { [I,{},<*v1,v2,c1,c2*>] where I is Element of Segm 15, v1, v2 is Element of SCM-Data-Loc , c1, c2 is Element of INT : I in {9,10,11,12,13} } ; :: thesis: f is set

then ex I being Element of Segm 15 ex v1, v2 being Element of SCM-Data-Loc ex c1, c2 being Element of INT st
( [y,f] = [I,{},<*v1,v2,c1,c2*>] & I in {9,10,11,12,13} ) ;
hence f is FinSequence by XTUPLE_0:1; :: thesis: verum

end;

end;