let X be set ; :: thesis: for f being PartFunc of REAL,REAL holds

( f | X is Lipschitzian iff ex r being Real st

( 0 < r & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds

|.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).| ) ) )

let f be PartFunc of REAL,REAL; :: thesis: ( f | X is Lipschitzian iff ex r being Real st

( 0 < r & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds

|.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).| ) ) )

thus ( f | X is Lipschitzian implies ex r being Real st

( 0 < r & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds

|.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).| ) ) ) :: thesis: ( ex r being Real st

( 0 < r & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds

|.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).| ) ) implies f | X is Lipschitzian )

A5: for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds

|.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).| ; :: thesis: f | X is Lipschitzian

take r ; :: according to FCONT_1:def 3 :: thesis: ( 0 < r & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds

|.(((f | X) . x1) - ((f | X) . x2)).| <= r * |.(x1 - x2).| ) )

thus 0 < r by A4; :: thesis: for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds

|.(((f | X) . x1) - ((f | X) . x2)).| <= r * |.(x1 - x2).|

let x1, x2 be Real; :: thesis: ( x1 in dom (f | X) & x2 in dom (f | X) implies |.(((f | X) . x1) - ((f | X) . x2)).| <= r * |.(x1 - x2).| )

assume A6: ( x1 in dom (f | X) & x2 in dom (f | X) ) ; :: thesis: |.(((f | X) . x1) - ((f | X) . x2)).| <= r * |.(x1 - x2).|

then ( (f | X) . x1 = f . x1 & (f | X) . x2 = f . x2 ) by FUNCT_1:47;

hence |.(((f | X) . x1) - ((f | X) . x2)).| <= r * |.(x1 - x2).| by A5, A6; :: thesis: verum

( f | X is Lipschitzian iff ex r being Real st

( 0 < r & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds

|.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).| ) ) )

let f be PartFunc of REAL,REAL; :: thesis: ( f | X is Lipschitzian iff ex r being Real st

( 0 < r & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds

|.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).| ) ) )

thus ( f | X is Lipschitzian implies ex r being Real st

( 0 < r & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds

|.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).| ) ) ) :: thesis: ( ex r being Real st

( 0 < r & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds

|.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).| ) ) implies f | X is Lipschitzian )

proof

given r being Real such that A4:
0 < r
and
given r being Real such that A1:
0 < r
and

A2: for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds

|.(((f | X) . x1) - ((f | X) . x2)).| <= r * |.(x1 - x2).| ; :: according to FCONT_1:def 3 :: thesis: ex r being Real st

( 0 < r & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds

|.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).| ) )

take r ; :: thesis: ( 0 < r & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds

|.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).| ) )

thus 0 < r by A1; :: thesis: for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds

|.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).|

let x1, x2 be Real; :: thesis: ( x1 in dom (f | X) & x2 in dom (f | X) implies |.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).| )

assume A3: ( x1 in dom (f | X) & x2 in dom (f | X) ) ; :: thesis: |.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).|

then ( (f | X) . x1 = f . x1 & (f | X) . x2 = f . x2 ) by FUNCT_1:47;

hence |.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).| by A2, A3; :: thesis: verum

end;A2: for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds

|.(((f | X) . x1) - ((f | X) . x2)).| <= r * |.(x1 - x2).| ; :: according to FCONT_1:def 3 :: thesis: ex r being Real st

( 0 < r & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds

|.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).| ) )

take r ; :: thesis: ( 0 < r & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds

|.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).| ) )

thus 0 < r by A1; :: thesis: for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds

|.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).|

let x1, x2 be Real; :: thesis: ( x1 in dom (f | X) & x2 in dom (f | X) implies |.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).| )

assume A3: ( x1 in dom (f | X) & x2 in dom (f | X) ) ; :: thesis: |.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).|

then ( (f | X) . x1 = f . x1 & (f | X) . x2 = f . x2 ) by FUNCT_1:47;

hence |.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).| by A2, A3; :: thesis: verum

A5: for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds

|.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).| ; :: thesis: f | X is Lipschitzian

take r ; :: according to FCONT_1:def 3 :: thesis: ( 0 < r & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds

|.(((f | X) . x1) - ((f | X) . x2)).| <= r * |.(x1 - x2).| ) )

thus 0 < r by A4; :: thesis: for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) holds

|.(((f | X) . x1) - ((f | X) . x2)).| <= r * |.(x1 - x2).|

let x1, x2 be Real; :: thesis: ( x1 in dom (f | X) & x2 in dom (f | X) implies |.(((f | X) . x1) - ((f | X) . x2)).| <= r * |.(x1 - x2).| )

assume A6: ( x1 in dom (f | X) & x2 in dom (f | X) ) ; :: thesis: |.(((f | X) . x1) - ((f | X) . x2)).| <= r * |.(x1 - x2).|

then ( (f | X) . x1 = f . x1 & (f | X) . x2 = f . x2 ) by FUNCT_1:47;

hence |.(((f | X) . x1) - ((f | X) . x2)).| <= r * |.(x1 - x2).| by A5, A6; :: thesis: verum