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

per cases ( [y,f] in (({[0,{},{}]} \/ { [I,{},<*a,b*>] where I is Element of Segm 8, a, b is Element of SCM-Data-Loc : I in {1,2,3,4} } ) \/ { [6,<*i*>,{}] where i is Nat : verum } ) \/ { [7,<*i*>,<*a*>] where i is Nat, a is Element of SCM-Data-Loc : verum } or [y,f] in { [5,{},<*a,r*>] where a is Element of SCM-Data-Loc , r is Element of S : verum } ) by A1, XBOOLE_0:def 3;

suppose A2: [y,f] in (({[0,{},{}]} \/ { [I,{},<*a,b*>] where I is Element of Segm 8, a, b is Element of SCM-Data-Loc : I in {1,2,3,4} } ) \/ { [6,<*i*>,{}] where i is Nat : verum } ) \/ { [7,<*i*>,<*a*>] where i is Nat, a is Element of SCM-Data-Loc : verum } ; :: thesis: f is set

per cases ( [y,f] in ({[0,{},{}]} \/ { [I,{},<*a,b*>] where I is Element of Segm 8, a, b is Element of SCM-Data-Loc : I in {1,2,3,4} } ) \/ { [6,<*i*>,{}] where i is Nat : verum } or [y,f] in { [7,<*i*>,<*a*>] where i is Nat, a is Element of SCM-Data-Loc : verum } ) by A2, XBOOLE_0:def 3;

suppose A3: [y,f] in ({[0,{},{}]} \/ { [I,{},<*a,b*>] where I is Element of Segm 8, a, b is Element of SCM-Data-Loc : I in {1,2,3,4} } ) \/ { [6,<*i*>,{}] where i is Nat : verum } ; :: thesis: f is set

per cases ( [y,f] in {[0,{},{}]} \/ { [I,{},<*a,b*>] where I is Element of Segm 8, a, b is Element of SCM-Data-Loc : I in {1,2,3,4} } or [y,f] in { [6,<*i*>,{}] where i is Nat : verum } ) by A3, XBOOLE_0:def 3;

suppose A4: [y,f] in {[0,{},{}]} \/ { [I,{},<*a,b*>] where I is Element of Segm 8, a, b is Element of SCM-Data-Loc : I in {1,2,3,4} } ; :: thesis: f is set

per cases ( [y,f] in {[0,{},{}]} or [y,f] in { [I,{},<*a,b*>] where I is Element of Segm 8, a, b is Element of SCM-Data-Loc : I in {1,2,3,4} } ) by A4, XBOOLE_0:def 3;

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

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

end;

suppose [y,f] in { [I,{},<*a,b*>] where I is Element of Segm 8, a, b is Element of SCM-Data-Loc : I in {1,2,3,4} } ; :: thesis: f is set

then consider I being Element of Segm 8, a, b being Element of SCM-Data-Loc such that
A5: ( [y,f] = [I,{},<*a,b*>] & I in {1,2,3,4} ) ;
f = <*a,b*> by A5, XTUPLE_0:1;
hence f is FinSequence ; :: thesis: verum

end;

end;

end;

suppose [y,f] in { [6,<*i*>,{}] where i is Nat : verum } ; :: thesis: f is set

then consider i being Nat such that
A6: [y,f] = [6,<*i*>,{}] ;
f = {} by A6, XTUPLE_0:1;
hence f is FinSequence ; :: thesis: verum

end;

end;

end;

suppose [y,f] in { [7,<*i*>,<*a*>] where i is Nat, a is Element of SCM-Data-Loc : verum } ; :: thesis: f is set

then consider i being Nat, a being Element of SCM-Data-Loc such that
A7: [y,f] = [7,<*i*>,<*a*>] ;
f = <*a*> by A7, XTUPLE_0:1;
hence f is FinSequence ; :: thesis: verum

end;

end;

end;

suppose [y,f] in { [5,{},<*a,r*>] where a is Element of SCM-Data-Loc , r is Element of S : verum } ; :: thesis: f is set

then consider a being Element of SCM-Data-Loc , r being Element of S such that
A8: [y,f] = [5,{},<*a,r*>] ;
f = <*a,r*> by A8, XTUPLE_0:1;
hence f is FinSequence ; :: thesis: verum

end;

end;