defpred S_{1}[ object , object ] means $2 = ((h . $1) + (g . $1)) - ((h . $1) * (g . $1));

A1: for x, y1, y2 being object st x in C & S_{1}[x,y1] & S_{1}[x,y2] holds

y1 = y2 ;

A2: for x, y being object st x in C & S_{1}[x,y] holds

y in REAL by XREAL_0:def 1;

consider IT being PartFunc of C,REAL such that

A3: ( ( for x being object holds

( x in dom IT iff ( x in C & ex y being object st S_{1}[x,y] ) ) ) & ( for x being object st x in dom IT holds

S_{1}[x,IT . x] ) )
from PARTFUN1:sch 2(A2, A1);

for x being object st x in C holds

x in dom IT

then A5: dom IT = C by XBOOLE_0:def 10;

then A6: for c being Element of C holds IT . c = ((h . c) + (g . c)) - ((h . c) * (g . c)) by A3;

for y being object st y in rng IT holds

y in [.0,1.]

then IT is Membership_Func of C by A5, FUNCT_2:def 1, RELAT_1:def 19;

hence ex b_{1} being Membership_Func of C st

for c being Element of C holds b_{1} . c = ((h . c) + (g . c)) - ((h . c) * (g . c))
by A6; :: thesis: verum

A1: for x, y1, y2 being object st x in C & S

y1 = y2 ;

A2: for x, y being object st x in C & S

y in REAL by XREAL_0:def 1;

consider IT being PartFunc of C,REAL such that

A3: ( ( for x being object holds

( x in dom IT iff ( x in C & ex y being object st S

S

for x being object st x in C holds

x in dom IT

proof

then
C c= dom IT
by TARSKI:def 3;
let x be object ; :: thesis: ( x in C implies x in dom IT )

A4: ex y being set st y = ((h . x) + (g . x)) - ((h . x) * (g . x)) ;

assume x in C ; :: thesis: x in dom IT

hence x in dom IT by A3, A4; :: thesis: verum

end;A4: ex y being set st y = ((h . x) + (g . x)) - ((h . x) * (g . x)) ;

assume x in C ; :: thesis: x in dom IT

hence x in dom IT by A3, A4; :: thesis: verum

then A5: dom IT = C by XBOOLE_0:def 10;

then A6: for c being Element of C holds IT . c = ((h . c) + (g . c)) - ((h . c) * (g . c)) by A3;

for y being object st y in rng IT holds

y in [.0,1.]

proof

then
rng IT c= [.0,1.]
by TARSKI:def 3;
reconsider A = [.0,jj.] as non empty closed_interval Subset of REAL by MEASURE5:14;

let y be object ; :: thesis: ( y in rng IT implies y in [.0,1.] )

assume y in rng IT ; :: thesis: y in [.0,1.]

then consider x being Element of C such that

A7: x in dom IT and

A8: y = IT . x by PARTFUN1:3;

0 <= (1_minus h) . x by Th1;

then A9: 0 <= 1 - (h . x) by FUZZY_1:def 5;

(1_minus g) . x <= 1 by Th1;

then A10: 1 - (g . x) <= 1 by FUZZY_1:def 5;

(1_minus h) . x <= 1 by Th1;

then 1 - (h . x) <= 1 by FUZZY_1:def 5;

then (1 - (h . x)) * (1 - (g . x)) <= 1 by A9, A10, XREAL_1:160;

then 1 - ((1 - (h . x)) * (1 - (g . x))) >= 1 - 1 by XREAL_1:10;

then 0 <= ((h . x) + (g . x)) - ((h . x) * (g . x)) ;

then A11: 0 <= IT . x by A3, A7;

A12: A = [.(lower_bound A),(upper_bound A).] by INTEGRA1:4;

then A13: 1 = upper_bound A by INTEGRA1:5;

0 <= (1_minus g) . x by Th1;

then 0 <= 1 - (g . x) by FUZZY_1:def 5;

then 0 <= (1 - (h . x)) * (1 - (g . x)) by A9, XREAL_1:127;

then 1 - 0 >= 1 - ((1 - (h . x)) * (1 - (g . x))) by XREAL_1:10;

then ((h . x) + (g . x)) - ((h . x) * (g . x)) <= 1 ;

then A14: IT . x <= 1 by A3, A7;

0 = lower_bound A by A12, INTEGRA1:5;

hence y in [.0,1.] by A8, A13, A11, A14, INTEGRA2:1; :: thesis: verum

end;let y be object ; :: thesis: ( y in rng IT implies y in [.0,1.] )

assume y in rng IT ; :: thesis: y in [.0,1.]

then consider x being Element of C such that

A7: x in dom IT and

A8: y = IT . x by PARTFUN1:3;

0 <= (1_minus h) . x by Th1;

then A9: 0 <= 1 - (h . x) by FUZZY_1:def 5;

(1_minus g) . x <= 1 by Th1;

then A10: 1 - (g . x) <= 1 by FUZZY_1:def 5;

(1_minus h) . x <= 1 by Th1;

then 1 - (h . x) <= 1 by FUZZY_1:def 5;

then (1 - (h . x)) * (1 - (g . x)) <= 1 by A9, A10, XREAL_1:160;

then 1 - ((1 - (h . x)) * (1 - (g . x))) >= 1 - 1 by XREAL_1:10;

then 0 <= ((h . x) + (g . x)) - ((h . x) * (g . x)) ;

then A11: 0 <= IT . x by A3, A7;

A12: A = [.(lower_bound A),(upper_bound A).] by INTEGRA1:4;

then A13: 1 = upper_bound A by INTEGRA1:5;

0 <= (1_minus g) . x by Th1;

then 0 <= 1 - (g . x) by FUZZY_1:def 5;

then 0 <= (1 - (h . x)) * (1 - (g . x)) by A9, XREAL_1:127;

then 1 - 0 >= 1 - ((1 - (h . x)) * (1 - (g . x))) by XREAL_1:10;

then ((h . x) + (g . x)) - ((h . x) * (g . x)) <= 1 ;

then A14: IT . x <= 1 by A3, A7;

0 = lower_bound A by A12, INTEGRA1:5;

hence y in [.0,1.] by A8, A13, A11, A14, INTEGRA2:1; :: thesis: verum

then IT is Membership_Func of C by A5, FUNCT_2:def 1, RELAT_1:def 19;

hence ex b

for c being Element of C holds b