let X be non empty set ; :: thesis: for f being PartFunc of [:X,X:],REAL

for a being Real st low_toler (f,a) is_reflexive_in X & f is symmetric holds

(low_toler (f,a)) [*] is Equivalence_Relation of X

let f be PartFunc of [:X,X:],REAL; :: thesis: for a being Real st low_toler (f,a) is_reflexive_in X & f is symmetric holds

(low_toler (f,a)) [*] is Equivalence_Relation of X

let a be Real; :: thesis: ( low_toler (f,a) is_reflexive_in X & f is symmetric implies (low_toler (f,a)) [*] is Equivalence_Relation of X )

assume that

A1: low_toler (f,a) is_reflexive_in X and

A2: f is symmetric ; :: thesis: (low_toler (f,a)) [*] is Equivalence_Relation of X

hence (low_toler (f,a)) [*] is Equivalence_Relation of X by A1, Th9; :: thesis: verum

for a being Real st low_toler (f,a) is_reflexive_in X & f is symmetric holds

(low_toler (f,a)) [*] is Equivalence_Relation of X

let f be PartFunc of [:X,X:],REAL; :: thesis: for a being Real st low_toler (f,a) is_reflexive_in X & f is symmetric holds

(low_toler (f,a)) [*] is Equivalence_Relation of X

let a be Real; :: thesis: ( low_toler (f,a) is_reflexive_in X & f is symmetric implies (low_toler (f,a)) [*] is Equivalence_Relation of X )

assume that

A1: low_toler (f,a) is_reflexive_in X and

A2: f is symmetric ; :: thesis: (low_toler (f,a)) [*] is Equivalence_Relation of X

now :: thesis: for x, y being object st x in X & y in X & [x,y] in low_toler (f,a) holds

[y,x] in low_toler (f,a)

then
low_toler (f,a) is_symmetric_in X
by RELAT_2:def 3;[y,x] in low_toler (f,a)

let x, y be object ; :: thesis: ( x in X & y in X & [x,y] in low_toler (f,a) implies [y,x] in low_toler (f,a) )

assume that

A3: ( x in X & y in X ) and

A4: [x,y] in low_toler (f,a) ; :: thesis: [y,x] in low_toler (f,a)

reconsider x1 = x, y1 = y as Element of X by A3;

f . (x1,y1) <= a by A4, Def3;

then f . (y1,x1) <= a by A2, METRIC_1:def 4;

hence [y,x] in low_toler (f,a) by Def3; :: thesis: verum

end;assume that

A3: ( x in X & y in X ) and

A4: [x,y] in low_toler (f,a) ; :: thesis: [y,x] in low_toler (f,a)

reconsider x1 = x, y1 = y as Element of X by A3;

f . (x1,y1) <= a by A4, Def3;

then f . (y1,x1) <= a by A2, METRIC_1:def 4;

hence [y,x] in low_toler (f,a) by Def3; :: thesis: verum

hence (low_toler (f,a)) [*] is Equivalence_Relation of X by A1, Th9; :: thesis: verum